In this paper we obtain the necessary and sufficient conditions for embedding results of different function classes. The main result is a criterion for embedding theorems for the so-called generalized Weyl-Nikol'skii class and the generalized Lipschitz class. To define the Weyl-Nikol'skii class, we use the concept of a $(\lambda,\beta)$-derivative, which is a generalization of the derivative in the sense of Weyl. As corollaries, we give estimates of norms and moduli of smoothness of transformed Fourier series.
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\begin{center}\Large{
Embedding theorems of function classes} \end{center}


\begin{center}\large{ B. Simonov, S. Tikhonov
}
\end{center}

\vspace{10mm}
\begin{abstract}
In this paper we obtain the necessary and sufficient conditions for embedding results of different
function classes. The main result is a criterion for embedding theorems for the so-called
generalized Weyl-Nikol'skii class and the generalized Lipschitz class. To define the
Weyl-Nikol'skii class, we use the concept of a $(\lambda,\beta)$-derivative, which is a
generalization of the derivative in the sense of Weyl. As corollaries, we give estimates of norms
and moduli of smoothness of transformed Fourier series.
\end{abstract}

\vspace{10mm}

Keywords: Embedding theorems, Lipschitz class, Weyl-Nikol'skii class,
$(\lambda,\beta)$-derivative, Moduli of smoothness of fractional order.

AMS: 46E35, 26A16, 26A33, 42A45.

\newpage

\section{Introduction}
 {\sf History of the question.} One of the main problems  of  constructive
approximation\footnote{The concept became well known through Bersntein's paper \cite[V. 2,
p.295-300; p. 349-360]{bern1}.} is in finding a relationship between differential properties of a
function  and its structural or construc\-ti\-ve characteristics. This topic started to develop
more than a century ago and in many cases the research was conducted as follows; authors
considered a given functional class and by investigating the properties of its elements obtained
embedding theorems with other functional classes. We recommend the articles by A. Pinkus
\cite{pin} and V. V. Zhuk and G. I. Natanson \cite{juk}  for a historical review.

The following three classical results gave rise to development of new areas within the
approximation theory:
\begin{align*}
&{\rm (A)}&
 \qquad f^{(r)}\in \textnormal{Lip} \,\alpha   \quad
&\Longleftrightarrow
 \qquad
E_n(f)= O\left(\frac{1}{n^{r+\alpha}}\right)& \qquad
&(0<\alpha<1, r\in \mathbf{Z}_+),\\
&{\rm (B)}&
 \qquad f^{(r)}\in \textnormal{Lip} \,\alpha
 \quad
&\Longleftrightarrow
 \qquad
\omega_{r+1}\Bigl(f,\frac{1}{n}\Bigr) = O \left(\frac{1}{n^{r+\alpha}}\right)& \qquad
&(0<\alpha<1, r\in \mathbf{Z}_+),
\\
&{\rm (C)}& \qquad f\in \textnormal{Lip}\, \alpha
 \quad
&\,\,\Longrightarrow
 \qquad
\widetilde{f}\in \textnormal{Lip} \,\alpha& \qquad &(0<\alpha<1).
\end{align*}

{\it Result} (A) was proved\footnote{See \cite{bern3} for a detailed review of the question before
the 30-ths XX century.}
 by
D. Jackson (1911, \cite{djek}) in the necessity part,  and by S. Bernstein (1912, \cite{bern},
\cite{bern2}) and Ch. de la Vall{\'e}e-Poussin (1919, \cite{vallee}) in the sufficiency part.

The theorems of this type are called direct and inverse theorems for trigonometric
appro\-xi\-ma\-ti\-on. Direct theorems for $L_p, 1\le p \le \infty$ (see the review \cite{juk})
are written as follows:
\begin{eqnarray}
E_n(f)_p&\le&  C(k)\, \,\omega_{k} \Bigl(f,\frac{1}{n+1}\Bigr)_p,
 \qquad\,\,\,\, k, n\in \mathbf{N},
\label{a1}\\
E_n(f)_p&\le&   \frac{C(k)}{n^r} \,\,
\omega_{k} \Bigl(f^{(r)},\frac{1}{n+1}
\Bigr)_p,  \quad\,\, k, n, r \in \mathbf{N}.
\label{a2}
\end{eqnarray}
Inverse theorems for $L_p, 1\le p \le \infty$ (see the review
\cite{juk}):
\begin{equation} \label{a3}
\omega_{k} \Bigl(f,\frac{1}{n+1}\Bigr)_p \le  \frac{C(k)}{(n+1)^k}
\sum\limits_{\nu=0}^n (\nu+1)^{k-1} E_\nu(f)_p ,
 \quad k, n\in \mathbf{N},
\end{equation}
\begin{equation} \label{a4}
\omega_{k} \Bigl(f^{(r)},\frac{1}{n+1}\Bigr)_p \le
C(k) \left(
\frac{1}{(n+1)^k}
\sum\limits_{\nu=0}^n (\nu+1)^{k+r-1} E_\nu(f)_p +
\sum\limits_{\nu=n+1}^\infty (\nu+1)^{r-1} E_\nu(f)_p
\right),  \,\, k, n\in \mathbf{N}.
\end{equation}
Here and further, the best trigonometric  approximation $E_n(f)_p$ and the modulus of smoothness
$\omega_k(f,\delta)_p$ are  defined by
$$
E_n(f)_p=\min\left( \left\|f-T \right\|_p; T\in \mathbf{T}_n \right),
\qquad \mathbf{T}_n=span\left\{\cos  mx, \sin mx : |m|\le n  \right\}
$$
and
\begin{equation} \label{mod}
\omega_k\left(f,\delta\right)_p = \sup \limits_{|h|\le\delta} \left\| \triangle^k_h f (x)
\right\|_p,
\end{equation}
$$ \qquad \triangle^k_h f (x) = \triangle^{k-1}_h \left(
\triangle_h f (x)\right) \qquad\mbox{and} \qquad \triangle_h f (x)=f(x+h)-f(x)
$$
respectively.

Note that the theorems on existence of the $r$-th derivative of $f$ from a given space have been
initiated by Bernstein \cite{bern}. He proved that the condition $\sum\limits_{\nu=1}^{\infty}
\nu^{r-1} E_\nu(f)_\infty<\infty$ implies $f^{(r)}\in C$. Later, for $L_p\,(1\le p \le \infty)$,
the following results were obtained (see the review \cite{juk} and the paper by O.V. Besov
\cite{bes1}). For convenience, we write these results in terms of the Besov space $B_{p,\theta}^r$
and the Sobolev space $W_p^r$:
\begin{align*}
\qquad &B_{p,1}^r\subset W^r_p \subset B_{p,\infty}^r\qquad p=1,\infty, \qquad
 \\
\qquad &B_{p,p}^r\subset W^r_p \subset B_{p,2}^r \qquad 1<p\le 2, \qquad
\\
\qquad &B_{p,2}^r\subset W^r_p \subset B_{p,p}^r \qquad 2\le p<\infty.
\end{align*}

{\it Result} (B) was proved by A. Zygmund (1945, \cite{zygmu-smooth}). He was one of the first to
use the modulus of smoothness concept of integer order introduced by S. Bernstein in 1912
(\cite{bern}). At present, the moduli of smoothness properties are well-studied (\cite{joh},
\cite{juk}) and the result (B) follows from inequalities (see \cite[Chapters 2 and 6]{devore},
\cite{johnen}):
\begin{equation} \label{b1}
\omega_{k+r} \Bigl(f,\frac{1}{n}\Bigr)_p \le  \frac{C(k,r)}{n^r}
\omega_{k} \Bigl(f^{(r)},\frac{1}{n}\Bigr)_p,
 \qquad k, r, n \in \mathbf{N}
\end{equation}
\begin{equation} \label{b2}
\omega_{k} \Bigl(f^{(r)},\frac{1}{n}\Bigr)_p \le  C(k,r)
\sum\limits_{\nu=n+1}^\infty \nu^{r-1}
\omega_{k+r} \Bigl(f,\frac{1}{\nu}\Bigr)_p,
 \,\, k, r, n \in \mathbf{N}.
\end{equation}
Comparing the last two inequalities and inequalities (\ref{a2}) and (\ref{a4}) we see that from
(\ref{b1}) and (\ref{b2}), using (\ref{a1}) and (\ref{a3}), it is easy to get (\ref{a2}) and
(\ref{a4}).


{\it Result} (C) was proved by I.I. Privalov (1919, \cite{priv}). The most complete version of the
inequality, from which embedding (C) follows,  was obtained by A. Zygmund (\cite{zygmu-smooth})
and N.K. Bary and S.B. Stechkin (\cite{ba-st}) and is the following one ($p=1,\infty$)
\begin{equation} \label{c1}
\omega_{k} \Bigl(\widetilde{f}^{(r)},\frac{1}{n}\Bigr)_p \le  C(k, r)
\left(
n^{-k}
\sum\limits_{\nu=1}^n \nu^{k+r-1}\omega_{k}\Bigl(f,\frac{1}{\nu}\Bigr)_p
+ \sum\limits_{\nu=n+1}^\infty \nu^{r-1}
\omega_{k} \Bigl(f,\frac{1}{\nu}\Bigr)_p
\right),
 \,\, k, n \in \mathbf{N}, r \in \mathbf{Z_+}.
\end{equation}
Finally, we note the paper by G.H. Hardy and J.S. Littlewood \cite{h-l} in which seemingly, for
the first time, some problems were formulated and solved in the same setting as in the present
paper. For the historical aspects of this approach see also \cite{og}.
\\
{\sf Embedding theorems for functional classes.} The results (A) - (C) as well as their
generalizations mentioned above can be written as the embedding theorems of the following
functional classes:
\begin{align*}
W^r_p &=\Bigl\{ f\in L_p: f^{(r)}\in L_p \Bigr\}, \\
\widetilde{W}^r_p &=\left\{ f\in L_p:\widetilde{f}^{(r)}\in L_p \right\}, \\
{W}^r_pH_\alpha[\varphi] &=\Bigl\{ f\in W^r_p:
\omega_\alpha \left(f^{(r)},\delta \right)_p = O[\varphi(\delta)] \Bigr\}, \\
\widetilde{W}^r_pH_\alpha[\phi] &=\left\{ f\in \widetilde{W}^r_p:
\omega_\alpha \left(\widetilde{f}^{(r)},
\delta \right)_p = O[\phi(\delta)] \right\}, \\
{W}^r _pE[\xi] &=\Bigl\{ f\in W^r_p:
E_n \left(f^{(r)}\right)_p = O[\xi(1/n)] \Bigr\}.
\end{align*}
We will study more general classes  for which  $W^r_p$,
$\widetilde{W}^r_p$, ${W}^r_pH_\alpha[\varphi]$,
$\widetilde{W}^r_pH_\alpha[\phi]$, ${W}^r E_p[\xi]$ are  particular cases.
\\
{\sf Transformed Fourier series}. Let $L_p=L_p[0, 2\pi] \,\,(1\le p< \infty)$ be a space of
$2\pi$-periodic  measurable functions for  which $|f|^p$ is integrable, and $L_\infty\equiv C[0,
2\pi]$ be the space of $2\pi$-periodic continuous functions with
$\|f\|_\infty=\max\limits\left\{|f(x)|, 0\le x\le 2\pi\right\}.$
\\
Let summable function $f(x)$ have the Fourier series
\begin{equation}\label{fourier}
f(x) \sim \sigma(f):= \frac{a_0 (f)}{2}+
\sum\limits_{\nu=1}^\infty
\left( a_\nu(f) \cos \nu x + b_\nu(f) \sin \nu x\right)
\equiv
\sum\limits_{\nu=0}^\infty A_\nu(f,x).
\end{equation}
By {\it the transformed Fourier series}  of (\ref{fourier}) we mean the series
 $$\sigma(f,\lambda,\beta):=\sum\limits_{\nu=1}^\infty
\lambda_\nu
\left[ a_\nu \cos\left( \nu x +\frac{\pi\beta}{2}\right) + b_\nu \sin
\left( \nu x +\frac{\pi\beta}{2}\right) \right],$$
where  $\beta\in \mathbf{R}$ and
 $\lambda=\left\{\lambda_n\right\}$ is a given sequence of positive numbers.

Studies of the  transformed Fourier series  are naturally related to the problems of Fourier
multipliers theory (see \cite{berg}, \cite{nikol'skij}, \cite[Vol 1, Chapter III]{zygm}),
summability methods (see \cite[Chapter 1.2]{butzbook}, \cite[Vol 1, Chapter III]{zygm})
and\footnote{See also references to \cite[\S 13, Chapter II]{baritext}.} so-called  the fractional
Sobolev classes or the Weyl classes.

We call the class
\begin{equation*}
W_p^{\lambda, \beta} =\Bigl\{
f\in L_p :\quad \exists \,\,g
 \in L_p, \quad g(x) \sim  \sigma(f,\lambda,\beta) \Bigr\}
\end{equation*}
{\it the Weyl class}. It is because $\lambda_n = n^r, r>0$ and $\beta = r$ the class
  $W_p^{\lambda, \beta}$ coincides with the class $W^r_p$, which is defined
in the term of fractional derivatives $f^{(r)}$ in the Weyl sense (\cite[Vol. 2, Chapter
XII]{zygm}). In the case $\lambda_n = n^r, r>0$ and $\beta = r+1$ the class $W_p^{\lambda, \beta}$
coincides with the class $\widetilde{W}^r_p$. We call the function $g(x) \sim
\sigma(f,\lambda,\beta)$ the $(\lambda,\beta)$-derivative of the function $f(x)$ and denote it by
$f^{(\lambda,\beta)}(x)$.
\\
{\sf Generalized Weyl-Nikolskii class}. In the definition of this functional class we use {\it the
modulus of smoothness} concept $\omega_{\alpha} (f,\delta)_p$ of fractional\footnote{ The term
"fractional" can be found in earlier papers
 (\cite{butz} and \cite{tab}) which used this  definition.
As in the case of fractional derivatives, the positive number\, $\alpha$ that defines  the modulus
order is not necessarily rational.}  order of a function $f(x) \in L_p$, i.e.,
$$\omega_{\alpha} (f,\delta)_p
 =\sup_{|h|\le \delta }
 \left\|
\triangle_{h}^{\alpha} f(x) \right\|_p,$$
where
 $$
\triangle_{h}^{\alpha} f(x) =\sum\limits_{\nu=0}^\infty(-1)^{\nu} \binom{\alpha}{\nu}
f(x+(\alpha-\nu)h), \qquad \alpha>0$$ is the $\alpha$-th difference\footnote{As usual,
 $\binom{\beta}{\nu} =
\frac{\beta(\beta-1)\cdots(\beta-\nu+1)}{\nu!}$ for $\nu
> 1$, $\binom{\beta}{\nu}=\beta$ for $\nu = 1$, and
 $\binom{\beta}{\nu} = 1 $ for $\nu = 0$.}
 of a function $f$ with step
$h$ at the point $x$.
 It is clear that for $\alpha\in \mathbf{N}$ this definition is the same as
(\ref{mod}).
\\ Let
$\Phi_\alpha\,\,(\alpha>0)$  be the class of functions
 $\varphi(\delta)$, defined and non-negative on  $(0,\pi]$ such that

1. $\varphi(\delta)\to 0\,\,\,(\delta\to 0)$,

2. $\varphi(\delta)$ is non-decreasing,

3. $\delta^{-\alpha}\varphi(\delta)$ is non-increasing.
\\
For such  $\alpha>0, \varphi\in \Phi_\alpha$ and $\lambda=\left\{\lambda_n\right\}$ {\it the
generalized Weyl-Nikolskii class} is defined similarly to the classes
 ${W}^r_pH^\alpha[\varphi]$ and
$\widetilde{W}^r_pH^\alpha[\phi]$:
\begin{equation*}
W_p^{\lambda, \beta} H_\alpha[\varphi] =
\Bigl\{
f\in
W_p^{\lambda, \beta} :\,\,\, \omega_\alpha\left( f^{(\lambda,\beta)},\delta
\right)_p=
O\left[\varphi(\delta)\right], \delta \to +0
 \Bigr\}.
\end{equation*}
It is clear that if  $\lambda_n = n^r, r>0$ and $\beta = r$, then
 $W_p^{\lambda, \beta} H_\alpha[\varphi]\equiv  {W}^r_pH_\alpha[\varphi]$
and if $\lambda_n = n^r, r>0$ and $\beta = r+1$, then
 $W_p^{\lambda, \beta} H_\alpha[\varphi]\equiv
 \widetilde{W}^r_pH_\alpha[\varphi]$.

In the case $\lambda_n\equiv 1$ and $\beta=0$ the class $W_p^{\lambda, \beta} H_\alpha[\varphi]$
coincides with {\it the generalized Lipschitz class } $H_\alpha^\varphi$, i.e.,
\begin{equation*}
H_\alpha^p[\varphi]
=
\Bigl\{
f\in L_p : \omega_\alpha(f,\delta)_p=
O\left[\varphi(\delta)\right]\quad \delta \to +0 \Bigr\}.
\end{equation*}
In particular,
\begin{equation*}
\textnormal{Lip} \,(\gamma, L_p) \equiv H_1^p \,[\delta^\gamma]= \Bigl\{ f\in L_p :
\omega_1(f,\delta)_p= O\left[\delta^\gamma\right]\quad \delta \to +0 \Bigr\}.
\end{equation*}
{\sf The problem setting and the structure of the paper}. In this paper, we obtain embedding
theorems for the Weyl class $W_p^{\lambda, \beta}$, for the generalized Weyl-Nikolskii class
$W_p^{\lambda, \beta} H_\alpha[\varphi]$ and for the generalized Lipschitz class
$H_\gamma^p[\omega]$. We show a relation between the parameters\, $\alpha$ and $\gamma$ depending
on the behavior of the sequence $\{\lambda_n\}$ and on the metric $L_p$.


The remainder of the paper is organized as follows. In section 2 we present the main theorem.
Sections 3 and  4 contain the proofs of the sufficiency and necessity parts of the main theorem
respectively. In section 5 we  provide several corollaries. In particular, we describe the
difference in results for metrics
 $L_p, 1<p<\infty$  and  $L_p, p=1,\infty$.
The estimates  $\omega_\gamma(f^{(r)}, \delta)_p$ and $\omega_\gamma(\widetilde{f}^{(r)},
\delta)_p$ are written in terms of $\omega_\beta(f, \delta)_p$ for different values of $r, \gamma$
and $\beta$. The concluding remarks are given in section 6.


\section{Embedding theorems for generalized Lipschitz and \\
 Weyl-Nikolskii   classes}

For $\lambda=\left\{\lambda_n\right\}_{n\in \mathbf{N}}$ we define $\triangle\lambda_n:=\lambda_n-
\lambda_{n+1}$; $\triangle^2\lambda_n:=\triangle\left(\triangle\lambda_n\right)$.

\begin{thr}
 Let\, $\theta=\min (2,p)$,  $\alpha\in  \mathbf{R}_+,$
$\beta\in \mathbf{R}$, and $\lambda=\left\{\lambda_n\right\}$ be a non-decreasing sequence of
positive numbers. Let  $\rho$  be a non-negative number such that the sequence
$\left\{n^{-\rho}\lambda_n\right\}$ is non-increasing.
\\
{\bf I. }\,\, Let $1<p<\infty$. Then
\begin{eqnarray}
\label{1} H_{\alpha+\rho}^p[\omega] \subset W_p^{ \lambda,\beta} \quad &\Longleftrightarrow&
\,\,\, \sum\limits_{n=1}^\infty  \left(\lambda_{n+1}^\theta-\lambda_{n}^\theta\right)
\omega^\theta\left(\frac{1}{n}\right)\, < \,\infty,
\\
H_{\alpha+\rho}^p[\omega] \subset W_p^{ \lambda,\beta} H_\alpha[\varphi] \,\,
&\Longleftrightarrow& \biggl\{ n^{-\alpha\theta} \sum\limits_{\nu=1}^n \nu^{(\rho+\alpha)\theta}
\left(\nu^{-\rho\theta}\lambda_{\nu}^\theta- (\nu+1)^{-\rho\theta}\lambda_{\nu+1}^\theta \right)
\omega^\theta\left(\frac{1}{\nu}\right) \nonumber
\\
&+& \sum\limits_{\nu=n+2}^\infty \left(\lambda_{\nu+1}^\theta-\lambda_{\nu}^\theta\right)
\omega^\theta\left(\frac{1}{\nu}\right) +
\lambda_{n+1}^\theta
\omega^\theta\left(\frac{1}{n+1}\right) \biggr\}^\frac{1}{\theta} \nonumber
\\
&=& \label{2} \quad  O\left[\varphi\left(\frac{1}{n+1} \right)\right],
\\
\label{3} W_p^{\lambda,\beta}
 \subset H_{\alpha+\rho}^p[\omega]
\quad &\Longleftrightarrow& \quad \frac{1}{\lambda_{n}}= O \left[\omega\left(\frac{1}{n}
\right)\right],
\\
\label{4} W_p^{ \lambda,\beta} H_\alpha[\varphi]
 \subset H_{\alpha+\rho}^p[\omega]
\quad &\Longleftrightarrow& \quad \frac{\varphi\left(\frac{1}{n} \right)}{\lambda_{n}}= O
\left[\omega\left(\frac{1}{n} \right)\right].
\end{eqnarray}
{\bf II. } \,\,Let $p=1$ or $p=\infty$.
\\
\textnormal{(a)} If  $\triangle^2\lambda_n\ge 0$ or
$\triangle^2\lambda_n\le 0$, then
\begin{eqnarray}\label{5}
H_{\alpha+\rho}^p[\omega] \subset W_p^{ \lambda,\beta} &\Longleftrightarrow& | \cos
\frac{\beta\pi}{2} | \sum\limits_{n=1}^\infty  \left(\lambda_{n+1}-\lambda_{n}\right)
\omega\left(\frac{1}{n}\right) \nonumber
\\
&+& | \sin \frac{\beta\pi}{2} | \sum\limits_{n=1}^\infty  \lambda_{n}
\frac{\omega\left(\frac{1}{n}\right)}{n} \, < \,\infty;
\end{eqnarray}
and if, additionally, for some  $\tau > 0$ the following inequality holds,
$$\triangle^2\left(\frac{\lambda_n}{n^r}\right)\ge 0 \quad\mbox{with}\quad
r=\rho+\tau\, sign \left|\sin\frac{(\beta-\rho)\pi}{2}\right|,$$ then
\begin{eqnarray}\label{6}
H_{\alpha+r}^p[\omega] \subset W_p^{ \lambda,\beta} H_\alpha[\varphi] &\Longleftrightarrow&
n^{-\alpha} \sum\limits_{\nu=1}^n \nu^{r+\alpha} \left(\nu^{-r}\lambda_{\nu}-
(\nu+1)^{-r}\lambda_{\nu+1} \right) \omega\left(\frac{1}{\nu}\right) \nonumber
\\
&+& | \cos \frac{\beta\pi}{2} | \sum\limits_{\nu=n+2}^\infty
\left(\lambda_{\nu+1}-\lambda_{\nu}\right) \omega\left(\frac{1}{\nu}\right) \nonumber
\\
&+& | \sin \frac{\beta\pi}{2} | \sum\limits_{\nu=n+2}^\infty
 \lambda_\nu
\frac{\omega\left(\frac{1}{\nu}\right)}{\nu}+ \lambda_{n+1} \omega\left(\frac{1}{n+1}\right)
\nonumber
\\&=& \quad
O\left[\varphi\left(\frac{1}{n+1}\right)\right].
\end{eqnarray}
\\
\textnormal{(b)} If  for $\beta= 2k, \, k\in \mathbf{Z}$, the condition
$\triangle^2\left(1/\lambda_n\right)\ge 0$ holds, and for $\beta\ne 2k, \, k\in \mathbf{Z},$
conditions $\triangle^2\left(1/\lambda_n\right)\ge 0$ and $\sum\limits_{\nu=n+1}^\infty
\frac{1}{\nu\lambda_\nu} \le \frac{C}{\lambda_n}$ are fulfilled, then
\begin{equation}\label{7}
W_p^{\lambda,\beta} \subset
 H^p_{\alpha+\rho}[\omega]
\quad \Longleftrightarrow \,\, \frac{1}{\lambda_{n}}= O \left[\omega\left(\frac{1}{n}
\right)\right];
\end{equation}
and if, additionally, for some  $\tau > 0$ the following inequality holds,
$$
\triangle^2\left(\frac{n^r}{\lambda_n}\right)\ge 0 \quad\mbox{or}\quad
\triangle^2\left(\frac{n^r}{\lambda_n}\right)\le 0
 \quad\mbox{with}\quad
r=\rho+\tau\, sign \left|\sin\frac{(\beta-\rho)\pi}{2}\right|,$$
 then
\begin{equation}\label{8}
W_p^{ \lambda,\beta} H_\alpha[\varphi]
 \subset H_{\alpha+r}^p[\omega]
\quad \Longleftrightarrow \quad \frac{\varphi\left(\frac{1}{n} \right)}{\lambda_{n}}= O
\left[\omega\left(\frac{1}{n} \right)\right].
\end{equation}
\end{thr}


\section{Proof of sufficiency in Theorem 1.}

We will use the following notations.
\\
Let a function $f(x)\in L$ have Fourier series (\ref{fourier}). Then $S_n(f)$ denotes the $n$-th
partial sum of (\ref{fourier}), $V_n(f)$ denotes the de la Vall{\'e}e-Poussin mean and $K_n(x)$ is the
Fej{\'e}r kernel, i.e.,
$$ \quad S_n(f)=\sum\limits_{\nu=0}^n A_\nu(x), \quad V_n(f)=\frac{1}{n}\sum\limits_{\nu=n}^{2n-1}
S_\nu(f), \quad K_n(x) = \frac{1}{n+1} \sum\limits_{\nu=0}^n \left(
\frac{1}{2}+\sum\limits_{m=1}^\nu \cos m x \right).$$ The following lemmas play the central role
in the proof of Theorem 1.

\begin{lemma}  \label{my}
Let  $f(x)\in L_p,$ $1\le p\le \infty$, and $\alpha>0$.
Then
\begin{equation} \label{m1}
C_1(p,\alpha) \omega_\alpha\Bigl(f,\frac{1}{n}\Bigr)_p \le \left( n^{-\alpha}\left\|
V_n^{(\alpha)}(f,x)) \right\|_p + \left\|f(x)-V_n(f,x)\right\|_p \right) \le C_2(p,\alpha)
\omega_\alpha\Bigl(f,\frac{1}{n}\Bigr)_p.
\end{equation}
If $f(x)\in L_p,$ $1< p < \infty$, then
\begin{equation} \label{m2}
C_1(p,\alpha) \omega_\alpha\Bigl(f,\frac{1}{n}\Bigr)_p \le \left( n^{-\alpha}\left\|
S_n^{(\alpha)}(f,x)) \right\|_p + \left\|f(x)-S_n(f,x)\right\|_p \right) \le C_2(p,\alpha)
\omega_\alpha\Bigl(f,\frac{1}{n}\Bigr)_p.
\end{equation}
\end{lemma}
 {\em Proof of Lemma  \ref{my}}. The estimate from above
$\omega_\alpha\Bigl(f,\frac{1}{n}\Bigr)_p$ follows from the inequality (see \cite{butz})
$\omega_\alpha\Bigl(T_n,\frac{1}{n}\Bigr)_p \le C(p,\alpha) n^{-\alpha}\left\| T_n^{(\alpha)}
\right\|_p$, where $T_n$ is a trigonometric polynomial of order $n$. We get
\begin{eqnarray*}
\omega_\alpha\Bigl(f,\frac{1}{n}\Bigr)_p
&\le&  C(p,\alpha)
\left(
\omega_\alpha\Bigl(T_n,\frac{1}{n}\Bigr)_p
+\left\| f-T_n \right\|_p
\right)
\\
&\le&  C(p,\alpha)
\left(
n^{-\alpha}\left\| T_n^{(\alpha)} \right\|_p
+ \left\| f-T_n \right\|_p  \right).
\end{eqnarray*}
Now we will estimate $\omega_\alpha\Bigl(f,\frac{1}{n}\Bigr)_p$ from below.
 We need the generalized Nikol'skii-Stechkin inequality
(see \cite{tab}) $n^{-\alpha}\left\| T_n^{(\alpha)} \right\|_p \le  C(p,\alpha)
\omega_\alpha\Bigl(T_n,\frac{1}{n}\Bigr)_p$ and the generalized Jackson inequality (see
\cite{butz})
\begin{equation}\label{11}
 E_n(f)_p\le  C(\alpha) \,\,
\omega_{\alpha} \left(f,\frac{1}{n+1}\right)_p.
\end{equation}
It is well known that the Vall{\'e}e-Poussin mean is the nearly best approximant, i.e.,
\begin{equation}\label{20}
\left\|f- V_n(f)\right\|_p\le  C E_n(f)_p.
\end{equation}
Then,
\begin{eqnarray*}
&&n^{-\alpha}\left\| V_n^{(\alpha)}(f,x) \right\|_p
+ \left\|f(x)-V_n(f,x)\right\|_p
\le C(p,\alpha) \left(
\omega_\alpha\Bigl(V_n,\frac{1}{n}\Bigr)_p + E_n(f)_p \right)
\\
&\le& C(p,\alpha) \left(
\omega_\alpha\Bigl(f,\frac{1}{n}\Bigr)_p +
\omega_\alpha\Bigl(f-V_n,\frac{1}{n}\Bigr)_p \right)
\le C(p,\alpha) \omega_\alpha\Bigl(f,\frac{1}{n}\Bigr)_p,
\end{eqnarray*}
and (\ref{m1}) is proved.
Using
\begin{equation}\label{zv}
\left\|f- S_n(f)\right\|_p\le  C(p)\, E_n(f)_p
\end{equation}
 for $1< p < \infty$, we will have (\ref{m2}) similarly.
This completes the proof of  Lemma \ref{my}.

We note that (\ref{m1}) and (\ref{m2}) are realization results
(see paper \cite{ditzian} by Z. Ditzian, V. H. Hristov, K. G.
Ivanov).


\begin{lemma}   \textnormal{(\cite{stechkin})}.
\label{4l22} Let  $f(x)\in L_p,$ $p=1,\infty$, and let
$\sum\limits_{k=1}^{\infty}k^{-1}E_k(f)_p<\infty$ be true. Then $\tilde{f}(x)\in L_p$ and
\begin{equation*}
E_n(\tilde{f})_p \le C \left( E_n(f)_p + \sum\limits_{k=n+1}^{\infty} k^{-1} E_k(f)_p  \right),
\qquad n\in  \mathbf{N}.
\end{equation*}
\end{lemma}

\begin{lemma} \label{my2}
Let $p=1,\infty$ and $\{\lambda_n\}$ be monotonic  concave (or convex) sequence. Let
\begin{eqnarray*}T_n(x)& = &\sum\limits_{\nu=0}^{n} a_\nu \cos \nu x +
b_\nu \sin \nu x, \\
T_n(\lambda, x)& = &\sum\limits_{\nu=0}^{n} \lambda_\nu\left(
a_\nu \cos \nu x + b_\nu \sin \nu x\right).
\end{eqnarray*}
Then for $M>N\ge 0$ one has
$$\left\| T_M(\lambda, x) - T_N(\lambda, x) \right\|_p \le \mu(M,N)
\left\| T_M(x) - T_N(x) \right\|_p,$$ where
$$ \mu(M,N) =
\begin{cases}
2M(\lambda_M- \lambda_{M-1})+
\lambda_{N+1} - (N+1)(\lambda_{N+2}- \lambda_{N+1}),
 &\text{if $\lambda_n \uparrow$
\textnormal{(}$n\uparrow$\textnormal{)}, $\triangle^2\lambda_n\ge 0$};\\
2\lambda_M+(N+1)(\lambda_{N+2}-\lambda_{N+1})-\lambda_{N+1},
 &\text{if $\lambda_n \uparrow$
\textnormal{(}$n\uparrow$\textnormal{)}, $\triangle^2\lambda_n\le 0$};\\
(N+1)(\lambda_{N+1}-\lambda_{N+2})+\lambda_{N+1},
 &\text{if $\lambda_n \downarrow$
\textnormal{(}$n\uparrow$\textnormal{)}, $\triangle^2\lambda_n\ge 0$}.
 \end{cases}
$$
\end{lemma}
{\em Proof of Lemma \ref{my2}}. Applying two times Abel's transformation we write
\begin{eqnarray*}
&\,& \left\| T_M(\lambda, x) - T_N(\lambda, x) \right\|_p = \frac{1}{\pi} \Bigl\|
\int\limits_{-\pi}^\pi \left(T_M - T_N\right) (x+u) \sum\limits_{\nu=N+1}^{M} \lambda_\nu \cos \nu
u \, du  \Bigr\|_p
\\
&=& \frac{1}{\pi} \Bigl\| \int\limits_{-\pi}^\pi \left(T_M - T_N\right) (x+u) \Bigl\{
\sum\limits_{\nu=N+1}^{M-2} \left(\lambda_{\nu}- 2 \lambda_{\nu+1} + \lambda_{\nu+2}\right)
(\nu+1) K_\nu(u)
\\
&+&
\left(\lambda_{N+2}- \lambda_{N+1}\right) (N+1) K_N(u) +
\left(\lambda_{M-1}- \lambda_{M}\right) M K_{M-1}(u)\Bigr\}du
+
\lambda_{M}
 \left(T_{M}-T_N\right) \Bigr\|_p
\\  &\le&
\left\| T_M(x) - T_N(x) \right\|_p
\Bigl\{
\sum\limits_{\nu=N+1}^{M-2}\left|\lambda_{\nu}- 2
\lambda_{\nu+1} + \lambda_{\nu+2}\right| (\nu+1) +
\left|\lambda_{M-1}- \lambda_{M}\right| M +
\lambda_{M} \Bigr\}
\\
&=:&
 \left\| T_M(x) - T_N(x) \right\|_p I(M,N).
\end{eqnarray*}
First we estimate $I(M,N)$  in the case $\lambda_n \uparrow$
($n\uparrow$), $\triangle^2\lambda_n\ge 0$. We have
\begin{eqnarray*}
I(M,N) &=&
\sum\limits_{\nu=N+1}^{M-2}
\left(\lambda_{\nu}- 2
\lambda_{\nu+1} + \lambda_{\nu+2}\right) (\nu+1) +
\left(\lambda_{M}- \lambda_{M-1}\right) M + \lambda_{M}
\\&=&
 -(N+1)\left(\lambda_{N+2}- \lambda_{N+1}\right) +
\left(\lambda_{N+1}- \lambda_{M-1}\right) +\lambda_{M}+
(2M-1)\left(\lambda_{M}- \lambda_{M-1}\right)
\\
&=&
 -(N+1)\left(\lambda_{N+2}- \lambda_{N+1}\right) +
\lambda_{N+1}+2M\left(\lambda_{M}- \lambda_{M-1}\right).
\end{eqnarray*}
If $\lambda_n \uparrow$ ($n\uparrow$), $\triangle^2\lambda_n\le 0$, then
$$
I(M,N) =
(N+1)\left(\lambda_{N+1}- \lambda_{N+2}\right) +
\left( \lambda_{M-1}-\lambda_{N+1}\right) +\lambda_{M}+
\left(\lambda_{M}- \lambda_{M-1}\right).
$$
Finally, if $\lambda_n \downarrow$
($n\uparrow$), $\triangle^2\lambda_n\ge 0$, then
$$
I(M,N) = (N+1)(\lambda_{N+1}-\lambda_{N+2})+\lambda_{N+1}.
$$
This completes the proof of  Lemma \ref{my2}.

\begin{lemma} \label{my222}
Suppose $p=1,\infty$. Let $ \,\,T_{2^n, \,2^{n+1}}(x) = \sum\limits_{\nu=2^n}^{2^{n+1}} \Bigl(
c_\nu \cos \nu x + d_\nu \sin \nu x \Bigr);$ then
\begin{equation}\label{trig}
C_1 \left\| \widetilde{T}_{2^n, \,2^{n+1}}(x) \right\|_p \le \left\| T_{2^n, \,2^{n+1}}(x)
\right\|_p \le C_2  \left\| \widetilde{T}_{2^n, \,2^{n+1}}(x) \right\|_p.
\end{equation}
\end{lemma}
{\em Proof of Lemma \ref{my222}}.
 We rewrite $T_{2^n, \,2^{n+1}}(x)$ in the following way
$$
T_{2^n, \,2^{n+1}}(x) = \sum\limits_{\nu=2^n}^{2^{n+1}}\frac{1}{\nu} \Bigl( \nu c_\nu \cos \nu x +
\nu d_\nu \sin \nu x \Bigr).
$$
Applying Lemma \ref{my2} and the Bernstein inequality we have
\begin{eqnarray*}
\left\| T_{2^n, \,2^{n+1}}(x) \right\|_p &\le& C \frac{1}{2^{n}} \left\|
\sum\limits_{\nu=2^n}^{2^{n+1}} \Bigl( \nu  c_\nu \cos \nu x + \nu d_\nu \sin \nu x \Bigr)
\right\|_p \\
&=& C  \frac{1}{2^{n}} \left\| \left( \sum\limits_{\nu=2^n}^{2^{n+1}}
 -d_\nu \cos \nu x + c_\nu \sin \nu x \right)'
\right\|_p \le C \left\| \widetilde{T}_{2^n, \,2^{n+1}}(x) \right\|_p.
\end{eqnarray*}
The same reasoning for $\widetilde{T}_{2^n, \,2^{n+1}}(x)$ works for the left-hand inequality of
(\ref{trig}). The proof is now complete.




\begin{center}
\bf{Sufficiency in (\ref{1}) - (\ref{8}).}
\end{center}

\textbf{I.} $1<p<\infty$. In this case, if $\lambda_n\equiv 1$, the Riesz inequality (\cite[V. 1,
p. 253]{zygm}) $\|\tilde{f}\|_p\le C(p) \|f\|_p$ implies
\begin{equation}\label{9}
\|f^{(\lambda,\beta)}\|_p \le C(p,\beta)\|f\|_p.
\end{equation}
Henceforth, $C(s,t,\cdots)$ will be positive constants that are dependent only on $s,t,\cdots$ and
may be different in different formulas.

Let the series in the  right part of (\ref{1}) be convergent and $f\in H^p_{\alpha+\rho}[\omega].$
We will use the following representation
$$\lambda_{2^n}^\theta = \lambda_{1}^\theta
+ \sum\limits_{\nu=2}^{n+1} \left( \lambda_{2^{\nu-1}}^\theta - \lambda_{2^{\nu-2}}^\theta
\right).$$ Applying the Minkowski's inequality, we get $\Bigl($ here and further
$\triangle_1:=A_1(f,x), \triangle_{n+2}:=\sum\limits_{\nu=2^n+1}^{2^{n+1}} A_\nu(f,x)$, where
$A_\nu(f,x)$ is from (\ref{fourier}) $ \Bigr)$

\begin{eqnarray} \label{AA}
I_1  &:=&   \left\{\int\limits_0^{2\pi}\left[
\sum\limits_{n=1}^{\infty}
 \lambda_{2^{n-1}}^2 \triangle^2_n\right]^\frac{p}{2}\,dx
\right\}^\frac{\theta}{p}
\nonumber
\\
&\le& C(p)\left(
 \lambda_{1}^\theta
\left\{ \int\limits_0^{2\pi}\left[ \sum\limits_{n=1}^{\infty} \triangle^2_n\right]^\frac{p}{2}\,dx
\right\}^\frac{\theta}{p} + \sum\limits_{s=2}^{\infty} \left( \lambda_{2^{s-1}}^\theta
-\lambda_{2^{s-2}}^\theta\right) \left\{ \int\limits_0^{2\pi}\left[ \sum\limits_{n=s}^{\infty}
\triangle^2_n\right]^\frac{p}{2}\,dx \right\}^\frac{\theta}{p} \right)^\frac{1}{\theta}.
\end{eqnarray}
By the Littlewood-Paley theorem  (\cite[V. II, p. 233]{zygm}) and (\ref{zv}), we write
\begin{equation}\label{zv1}
I_1\le C(p)\left\{
 \lambda_{1}^\theta
 \|f\|^\theta_p +
\sum\limits_{s=1}^{\infty} \left( \lambda_{2^{s}}^\theta -\lambda_{2^{s-1}}^\theta\right)
E_{2^{s-1}}^\theta(f)_p \right\}^\frac{1}{\theta}.
\end{equation}
Then, both the generalized Jackson inequality (\ref{11}) and   $f\in H^p_{\alpha+\rho}[\omega]$
imply $I_1<\infty.$ Thus, there exists a function  $g\in L_p$ with Fourier series
\begin{equation}\label{12}
\sum\limits_{n=1}^{\infty} \lambda_{2^{n-1}} \triangle_n,
\end{equation}
and also $\|g\|_p\le C(p) I_1$. We rewrite series  (\ref{12}) in the form of
 $\sum\limits_{n=1}^{\infty}
\gamma_{n} A_n(f,x)$, where $\gamma_{i}:=\lambda_i, \,i=1,2$ and $\gamma_{\nu}:= \lambda_{2^{n}}$
for $2^{n-1}+1\le \nu \le 2^{n} \,(n=2,3,\cdots).$
 Further, we write the series
\begin{equation}\label{13}
\sum\limits_{n=1}^{\infty}   \lambda_{n}A_n(f,x) =
\sum\limits_{n=1}^{\infty}   \gamma_n \Lambda_{n}A_n(f,x),
\end{equation}
where $\Lambda_{1}=\Lambda_{2}=1$, $\Lambda_{\nu} := \lambda_{\nu}/\gamma_{n} =
\lambda_{\nu}/\lambda_{2^{n}}$ for $2^{n-1}+1\le \nu \le 2^{n} \,(n=2,3,\cdots).$
 The sequence $\left\{\Lambda_{n}\right\}$ satisfies the conditions of the Marcinkiewicz
multiplier theorem (\cite[V.II, p. 232]{zygm}), i.e., series (\ref{13}) is the Fourier series of a
function $f^{(\lambda, 0)} \in L_p$, $\|f^{(\lambda, 0)}\|_p \le C(p) \|g\|_p.$ Then, inequalities
(\ref{11}), (\ref{9}) and  (\ref{zv1}) imply
\begin{eqnarray}\label{14}
\|f^{(\lambda,\beta)}\|_p &\le& C(p,\beta) \left\{
 \lambda_{1}^\theta
 \|f\|^\theta_p +
\sum\limits_{s=1}^{\infty}
E_{2^{s-1}}^\theta(f)_p
\sum\limits_{n=2^{s-1}}^{2^{s}-1}
\left( \lambda_{n+1}^\theta -\lambda_n^\theta\right)
  \right\}^\frac{1}{\theta}
\label{14}\\
&\le& C(p, \beta, \alpha, \rho) \left\{
 \lambda_{1}^\theta
 \|f\|^\theta_p +
\sum\limits_{n=1}^{\infty}
\left( \lambda_{n+1}^\theta -\lambda_n^\theta\right)
 \omega_{\alpha+\rho}^\theta \left(f,\frac{1}{n}\right)_p
 \right\}^\frac{1}{\theta},
\nonumber
\end{eqnarray}
i.e., the sufficiency in (\ref{1}) has been proved.

Let the inequality in (\ref{2}) hold, and $f\in
H^p_{\alpha+\rho}[\omega]$. Let us prove $f\in
W^{\lambda,\beta}_pH_{\alpha}[\varphi].$ First we estimate
$\omega_\alpha(f^{(\lambda,\beta)},\frac{1}{n})_p.$ By Lemma
\ref{my},
\begin{equation}\label{16}
\omega_\alpha\left(f^{(\lambda,\beta)},\frac{1}{n}\right)_p \le C(p,\alpha) \Bigl(
\|f^{(\lambda,\beta)} - S_n(f^{(\lambda,\beta)}) \|_p+ n^{-\alpha}
\|S^{(\alpha)}_n(f^{(\lambda,\beta)}) \|_p
 \Bigr).
\end{equation}
Using (\ref{14}) for the function $\left(f - S_n\right)$, we have ($[a]$ is the integer part of
$a$)
\begin{align} \label{17}
\|f^{(\lambda,\beta)} - S_n(f^{(\lambda,\beta)})\|_p \le C(p,
\beta, \alpha, \rho) \Biggl\{
 \lambda_{n+1}^\theta
\left\|f - S_n\right\|^\theta_p \,
&+ \,\,
E_{\left[\frac{n}{2}\right]}^\theta(f)_p
\sum\limits_{s=1}^{2n}
\left( \lambda_{s+1}^\theta -\lambda_s^\theta\right)
\nonumber
\\
&+        \,\,
\sum\limits_{s=n+1}^{\infty}
\left( \lambda_{s+1}^\theta -\lambda_s^\theta\right)
E_{\left[\frac{s}{2}\right]}^\theta(f)_p
  \Biggr\}^\frac{1}{\theta}
\nonumber
\\
\le              \, C(p, \beta, \alpha, \rho) \Biggl\{
 \lambda_{n+1}^\theta
\omega^\theta_{\alpha+\rho}
\left(f,\frac{1}{n}\right)_p
\,\,
&+  \,
\left.
\sum\limits_{\nu=n+1}^{\infty}
\left( \lambda_{\nu+1}^\theta -\lambda_\nu^\theta\right)
 \omega_{\alpha+\rho}^\theta \left(f,\frac{1}{\nu}\right)_p
 \right\}^\frac{1}{\theta}.
\end{align}

Further, we estimate the second item of (\ref{16}). Let $m$ be an integer such that $2^m\le n+1<
2^{m+1}.$ We will use here the representation
$$2^{-s\rho\theta}\lambda_{2^s}^\theta =
2^{-(m+1)\rho\theta} \lambda_{2^{m+1}}^\theta + \sum\limits_{\nu=s}^{m} \left(
2^{-\nu\rho\theta}\lambda_{2^\nu}^\theta - 2^{-(\nu+1)\rho\theta}\lambda_{2^{\nu+1}}^\theta
\right).$$ Then,  using Lemmas \ref{my}, \ref{my2}, we can follow the way of proof in
(\ref{AA})-(\ref{14}). Then, one can obtain
\begin{eqnarray} \label{18}
n^{-\alpha} \|S^{(\alpha)}_n\left(f^{(\lambda,\beta)}\right) \|_p
&\le&
 C(p, \beta, \alpha, \rho) \Biggl\{
  \lambda_{n+1}^\theta
\omega^\theta_{\alpha+\rho}(f,\frac{1}{n})_p
\nonumber
\\ &+&
n^{-\alpha\theta}
\sum\limits_{\nu=1}^{n}
\left(
\nu^{-\rho\theta}\lambda_{\nu}^\theta -
(\nu+1)^{-\rho\theta}\lambda_{\nu+1}^\theta
\right)
\nu^{(\rho+\alpha)\theta}
 \omega_{\alpha+\rho}^\theta \left(f,\frac{1}{\nu}\right)_p
 \Biggr\}^\frac{1}{\theta}.
\end{eqnarray}
We use (\ref{17}), (\ref{18}) and the right part of (\ref{2}) to obtain  $f\in
W^{\lambda,\beta}_pH_{\alpha}[\varphi].$

Now we prove that conditions $\frac{1}{\lambda_{n}} = O
\left[\omega\left(\frac{1}{n}\right)\right]$ and $\frac{\varphi\left(\frac{1}{n}
\right)}{\lambda_{n}}= O \left[\omega\left(\frac{1}{n} \right)\right]$ are sufficient for
$W^{\lambda,\beta}_p \subset H^p_{\alpha+\rho}[\omega]$ and $W^{\lambda,\beta}_p
H_{\alpha}[\varphi]
 \subset H^p_{\alpha+\rho}[\omega]$, respectively.

 From the properties of the sequence $\left\{\lambda_n\right\}$, using the
Littlewood-Paley and the Marcinkiewicz multiplier theorem, we get
\begin{eqnarray*}
 \omega_{\alpha+\rho}\left(f,\frac{1}{n}\right)_p
&\le& C(p, \alpha, \rho) \Bigl( \|f - S_{n}(f) \|_p +
n^{-(\alpha+\rho)} \|S^{(\alpha+\rho)}_{n}(f) \|_p \Bigr)
\\
&\le& C(p, \beta, \alpha, \rho) \Bigl( \lambda^{-1}_{n}
\|f^{(\lambda,\beta)} - S_{n}\left(f^{(\lambda,\beta)}\right) \|_p
+ \lambda^{-1}_{n} n^{-\alpha}
 \|S^{(\alpha)}_{n}(f^{(\lambda,\beta)}) \|_p
\Bigr).
\end{eqnarray*}
Then, by Lemma \ref{my}, the following inequalities are true
$$
\omega_{\alpha+\rho}\left(f,\frac{1}{n}\right)_p \le C(p, \beta,
\alpha, \rho) \lambda^{-1}_{n}
\omega_{\alpha}\left(f^{(\lambda,\beta)},\frac{1}{n}\right)_p \le
C(p, \beta, \alpha, \rho) \lambda^{-1}_{n}
\|f^{(\lambda,\beta)}\|_p.
$$
Thus, the first part implies sufficiency in (\ref{4}) and the second implies sufficiency in
(\ref{3}).

 {\bf II.} $p=1$ or $p=\infty$. Let the series in (\ref{5}) be convergent, and let $f\in
H^p_{\alpha+\rho}[\omega]$. Consider the series
\begin{eqnarray} \label{19}
\cos \frac{\pi\beta}{2} V_1(\lambda,f)& -&
\sin \frac{\pi\beta}{2} \widetilde{V_1}(\lambda,f)
 \\
&+&
\sum\limits_{n=1}^{\infty}
\left\{
\cos \frac{\pi\beta}{2}
\left(
 V_{2^n}(\lambda,f) -  V_{2^{n-1}}(\lambda,f) \right)
-  \sin \frac{\pi\beta}{2} \left( \widetilde{V_{2^n}}(\lambda,f) -
\widetilde{V_{2^{n-1}}}(\lambda,f) \right) \right\}, \nonumber
\end{eqnarray}
where $V_{1}(\lambda,f) := \lambda_1 A_1(f,x),$
$$V_{n}(\lambda,f)
:= \sigma(\lambda, V_n(f)) =
 \sum\limits_{m=1}^{n}
 \lambda_m A_m(f,x)
+
\sum\limits_{m=n+1}^{2 n-1}  \lambda_m\left(1-\frac{m-n}{n}\right)
 A_m(f,x) \,(n\ge 2).$$
Let $M>N>0$. From the inequality $\left\|f- V_n(f)\right\|_p\le  C E_n(f)_p$ and the Jackson
inequality (\ref{11}), and using the properties of  $\left\{\lambda_n\right\}$ and the outline of
Lemma \ref{my2}, we get
\begin{eqnarray} \label{21}
A
&:=&\left\|
 \sum\limits_{n=N}^{M}\left[ \cos \frac{\pi\beta}{2}
\left(
 V_{2^{n+1}}(\lambda,f) -  V_{2^{n}}(\lambda,f)
\right)
- \sin \frac{\pi\beta}{2}
\left(
\widetilde{V_{2^{n+1}}}(\lambda,f) - \widetilde{V_{2^{n}}}(\lambda,f) \right)
\right]
\right\|_p
\nonumber \\
&\le& \sum\limits_{n=N}^{M}\left[ | \cos \frac{\pi\beta}{2}| \left\|  V_{2^{n+1}}(f) -
V_{2^{n}}(f)  \right\|_p \left( \sum\limits_{m=2^{n}}^{2^{n+2}-1} |\triangle^2 \lambda_{m}| (m+1)
+ 2^{n+2} |\triangle \lambda_{2^{n+2}}| \right) \right.
\nonumber \\
&+&
\left. | \sin \frac{\pi\beta}{2}|
\left\|
\widetilde{V_{2^{n+1}}}(f) - \widetilde{V_{2^{n}}}(f)
 \right\|_p
\left( \sum\limits_{m=2^{n}}^{2^{n+2}-1} |\triangle^2 \lambda_{m}| (m+1) + 2^{n+2} |\triangle
\lambda_{2^{n+2}-1}| \right) \right]
\nonumber \\
&+&
|\cos \frac{\pi\beta}{2}|
\left\|
\sum\limits_{n=N}^{M}
 \lambda_{2^{n+2}}
\left(  V_{2^{n+1}} -  V_{2^n}\right)(f)  \right\|_p
+
|\sin \frac{\pi\beta}{2}|
\left\|
\sum\limits_{n=N}^{M}
 \lambda_{2^{n+2}}
\left(  \widetilde{V_{2^{n+1}}} - \widetilde{ V_{2^n}}
\right)(f)  \right\|_p
\nonumber \\
&\le&
C \Biggl\{
 \lambda_{2^{N}} \left(
|\cos \frac{\pi\beta}{2}| E_{2^{N}} (f)_p + |\sin \frac{\pi\beta}{2}| E_{2^{N}} (\tilde{f})_p
\right)
\nonumber \\
&+& \sum\limits_{n=2^{N}-1}^{\infty} \left( \lambda_{n+1} -  \lambda_{n}\right) \left( |\cos
\frac{\pi\beta}{2}|\,\, \omega_{\alpha+\rho}\left(f, \frac{1}{n}\right)_p + |\sin
\frac{\pi\beta}{2}|\,\, \omega_{\alpha+\rho}\left(\tilde{f}, \frac{1}{n}\right)_p \right)
\Biggr\}.
\end{eqnarray}
To complete the proof of the sufficiency part in (\ref{5}), we apply Lemma \ref{4l22}, inequality
(\ref{a3}) (see \cite{tab} for the case $k>0$),  and inequality  (\ref{11}). Then the convergence
of series in (\ref{5}) and $f\in H^p_{\alpha+\rho}[\omega]$ imply that there exists $\varphi\in
L_p$ such that series (\ref{19}) converges to $\varphi$ in $L_p$. Let us show that
$\sigma(\varphi) = \sigma(f^{(\lambda,\beta)}).$ If $F_n$ is the $n$-th partial sum of (\ref{19}),
then, say for cosine coefficients,
$$
a_n(\varphi) =
a_n(\varphi-F_{{N+n}})+a_n(F_{{N+n}}) =
a_n(\varphi-F_{{N+n}}) +
a_n(f^{(\lambda,\beta)}),
$$
and
$$a_n(\varphi-F_{{N+n}}) =  \frac{1}{\pi} \int\limits_{-\pi}^{\pi}
\left(\varphi- F_{{N+n}} \right)(x) \cos nx \, dx \le
C(p) \left\|\varphi- F_{{N+n}} \right\|_p \longrightarrow
\,0 \,\,(N\to \infty).
$$
This completes the proof of the sufficiency part of (\ref{5}).

Let $f\in H^p_{\alpha+r}[\omega]$ and the condition in the right part of (\ref{6}) hold. We will
estimate from above $\omega_\alpha\left(f^{(\lambda,\beta)},\frac{1}{n}\right)_p$. By Lemma
\ref{my},
$$
\omega_\alpha\left(f^{(\lambda,\beta)},\frac{1}{n}\right)_p \le C(\alpha)\left(
\left\|f^{(\lambda,\beta)} -V_n(f^{(\lambda,\beta)})\right\|_p+ n^{-\alpha}
\left\|V_n^{(\alpha)}(f^{(\lambda,\beta)})\right\|_p \right).$$
Let us show that
\begin{eqnarray}  \label{23}
\left\|f^{(\lambda,\beta)} -V_n(f^{(\lambda,\beta)})\right\|_p
&\le& C(\beta, \alpha, r)
 \left(  \lambda_{n}
\omega_{\alpha+r}\left( f, \frac{1}{n}\right)_p +|\cos
\frac{\pi\beta}{2}|\, \sum\limits_{\nu=n+1}^{\infty} \left(
\lambda_{\nu+1} -  \lambda_{\nu}\right) \omega_{\alpha+r}\left( f,
\frac{1}{\nu}\right)_p \right.\nonumber
\\
&+&
\left.
|\sin \frac{\pi\beta}{2}|\,
\sum\limits_{\nu=n+1}^{\infty}
\frac{\lambda_{\nu}}{\nu}
\omega_{\alpha+r}\left( f, \frac{1}{\nu}\right)_p
 \right).
\end{eqnarray}
We have already shown
 that  $A= \left\|
  \sum\limits_{n=N}^{M}
  \left(
 V_{2^{n+1}}(f^{(\lambda,\beta)}) -  V_{2^{n}}(f^{(\lambda,\beta)})
 \right)
\right\|_p.$ Then for  $2^m\le n<2^{m+1}$
\begin{eqnarray*}
\left\|f^{(\lambda,\beta)} -V_n(f^{(\lambda,\beta)})\right\|_p &\le& \left\|
 V_{n}(f^{(\lambda,\beta)}) - V_{2^{m+1}}(f^{(\lambda,\beta)})
\right\|_p + \left\| \sum\limits_{\nu=m+1}^{\infty} \left(
 V_{2^\nu}(f^{(\lambda,\beta)}) - V_{2^{\nu+1}}(f^{(\lambda,\beta)})\right) \right\|_p
\\&=:& I_1+I_2.
\end{eqnarray*}
By Lemma \ref{my222}, we have
\begin{eqnarray*}
I_1 &=& \left\| \cos \frac{\pi\beta}{2} \left( V_{n}(\lambda, f) - V_{2^{m+1}} (\lambda, f)
\right) - \sin \frac{\pi\beta}{2} \left( \widetilde{V_{n}}(\lambda, f) - \widetilde{V_{2^{m+1}}}
(\lambda, f) \right) \right\|_p
\\
&\le& \left\| V_{n}(\lambda, f) - V_{2^{m+1}}(\lambda, f)\right\|_p
+
\left\|\widetilde{V_{n}}(\lambda, f) - \widetilde{V_{2^{m+1}}} (\lambda, f) \right\|_p
 \le
 C \left\|
V_{n}(\lambda, f) - V_{2^{m+1}}(\lambda, f)\right\|_p
\end{eqnarray*}
and, by Lemma \ref{my2}, we get $ I_1 \le \lambda_{n} \omega_{\alpha+r}\left( f,
\frac{1}{n}\right)_p$.

Secondly
\begin{eqnarray*}
I_2 &\le& |\cos \frac{\pi\beta}{2}|
 \left\|
 \sum\limits_{\nu=m+1}^{\infty} \left( V_{2^\nu}(\lambda, f) - V_{2^{\nu+1}}
(\lambda, f) \right) \right\|_p + | \sin \frac{\pi\beta}{2}|
 \left\|
 \sum\limits_{\nu=m+1}^{\infty}
 \left( \widetilde{V_{2^\nu}}(\lambda, f) - \widetilde{V_{2^{\nu+1}}} (\lambda, f) \right)\right\|_p.
\end{eqnarray*}
As in (\ref{21}), we have
\begin{eqnarray*}
 &&\left\|
 \sum\limits_{\nu=N}^{M} \left( V_{2^{\nu+1}} - V_{2^{\nu}}\right)
(\lambda, f)  \right\|_p \le C\left( \lambda_{2^{N}} \omega_{\alpha+r}\left( f,
\frac{1}{2^{N}}\right)_p +
 \sum\limits_{\nu=2^{N}+1}^{\infty}
 \left( \lambda_{\nu+1} -  \lambda_{\nu}\right)
\omega_{\alpha+r}\left( f, \frac{1}{\nu}\right)_p
 \right)
 ;
 \\&&
 \left\|
 \sum\limits_{\nu=N}^{M} \left(
\widetilde{V_{2^\nu}}(\lambda, f) - \widetilde{V_{2^{\nu+1}}} (\lambda, f)
  \right) \right\|_p
  \le
   \left\|
 \sum\limits_{\nu=N}^{M}
 \lambda_{2^{\nu+2}}
 \left(
\widetilde{V_{2^{\nu}}}(f) - \widetilde{V_{2^{\nu+1}}}(f)
  \right) \right\|_p
  \\
&&+
 \sum\limits_{\nu=N}^{M}
 \left\|\widetilde{V_{2^{\nu}}}(f) - \widetilde{V_{2^{\nu+1}}}(f) \right\|_p
\left( \sum\limits_{m=2^{\nu}}^{2^{\nu+2}-1} |\triangle^2 \lambda_{m}| (m+1) + 2^{\nu+2}
|\triangle \lambda_{2^{\nu+2}}| \right) =: I_{21}+I_{22}.
\end{eqnarray*}
By Lemma \ref{4l22},
\begin{eqnarray*}
 I_{21}
  &\le& \lambda_{2^{N+1}}
  \left\| \widetilde{V_{2^{M+1}}}(f) - \widetilde{V_{2^N}}(f)  \right\|_p
  +
 \sum\limits_{\nu=N}^{M}  \left(\lambda_{2^{\nu+2}}-\lambda_{2^{\nu+1}}\right)
\left\| \widetilde{V_{2^{\nu+1}}}(f) - \widetilde{V_{2^{\nu}}}(f)  \right\|_p
  \\
 &\le& C\left(
 \lambda_{2^{N+1}}
 \sum\limits_{\nu=N}^{\infty} E_{2^\nu}(f)_p
  +
 \sum\limits_{\nu=N}^{\infty}  \left(\lambda_{2^{\nu+2}}-\lambda_{2^{\nu+1}}\right)
 \sum\limits_{s=\nu}^{\infty}  E_{2^s}(f)_p\right)
\le C
 \sum\limits_{\nu=2^N}^{\infty} \frac{\lambda_{\nu}}{\nu}
\omega_{\alpha+r}\left( f, \frac{1}{\nu}\right)_p,
 \\
 I_{22}
  &\le& C
 \sum\limits_{\nu=N}^{M}  \left(\lambda_{2^{\nu+3}}-\lambda_{2^{\nu-1}}\right)
 E_{2^\nu}(\tilde{f})_p
 \le C
 \sum\limits_{\nu=2^N}^{\infty} \frac{\lambda_{\nu}}{\nu}
\omega_{\alpha+r}\left( f, \frac{1}{\nu}\right)_p,
\end{eqnarray*}
and (\ref{23}) follows.

Repeating the argument in  (\ref{21}), we estimate
 $n^{-\alpha} \left\|V_n^{(\alpha)}(f^{(\lambda,\beta)})\right\|_p$.
By Lemma \ref{my2} and by inequalities (\ref{11}) and (\ref{20}), we write
\begin{equation}  \label{24}
\left\| V_n^{(\alpha)}(f^{(\lambda,\beta)}) \right\|_p \le
C(\beta, \alpha, r) \left(  n^\alpha \lambda_{n}
\omega_{\alpha+r}\left( f, \frac{1}{n}\right)_p +
\sum\limits_{\nu=1}^{n} \left(\frac{ \lambda_{\nu}}{\nu^r} -
\frac{ \lambda_{\nu+1}}{(\nu+1)^r} \right) \nu^{\alpha+r}
\omega_{\alpha+r}\left( f, \frac{1}{\nu}\right)_p
 \right).
\end{equation}
By means of (\ref{23}) and (\ref{24}) we get $f\in W_p^{ \lambda,\beta} H_\alpha[\varphi],$
because of the condition in (\ref{6}).

Let us prove (\ref{7}). Let $f\in W_p^{ \lambda,\beta}.$ To establish $f\in
H_{\alpha+\rho}^p(\omega)$, we shall first estimate $E_n(f)_p$ for the case $\sin
\frac{\pi\beta}{2}\ne 0$. Repeating the reasoning in (\ref{21}), by  (\ref{20}), we get ( $2^m\le
n<2^{m+1}$ )
\begin{eqnarray*}
E_n(f)_p &\le& C \sum\limits_{\nu=m}^{\infty} \frac{1}{\lambda_{2^\nu}} \left\|
\left(V_{[2^{\nu-1}]}- V_{2^\nu} \right) (f^{(\lambda,\beta)}) \right\|_p
\nonumber
\\
&\le&
 C
E_{[2^{m-1}]}(f^{(\lambda,\beta)})_p \sum\limits_{\nu=m}^{\infty} \frac{1}{\lambda_{2^\nu}} \le
C(\rho, p)
   \frac{\|f^{(\lambda,\beta)}\|_p }{\lambda_n}.
\end{eqnarray*}
If $\sin \frac{\pi\beta}{2}= 0$, it is easy to see that
$$E_n(f)_p \le
\frac{ C}{\lambda_n} E_{n}(f^{(\lambda,\beta)})_p \le
   \frac{C}{\lambda_n}\|f^{(\lambda,\beta)}\|_p . $$
Substituting the bound for $E_n(f)_p$ into (\ref{a3}) and using the fact that
$n^\rho\lambda_n^{-1}\uparrow$ ($n\uparrow$), we can write
$$
\omega_{\alpha+\rho}\left( f, \frac{1}{n}\right)_p
 \le C(\alpha,\rho) \frac{1}{n^{\alpha+\rho}}
\sum\limits_{\nu=0}^{n} \nu^{\alpha+\rho-1} E_{\nu-1}(f)_p \le
 \frac{C(\alpha,\rho)}{\lambda_n} = O
 \left[\omega\left(\frac{1}{n} \right)\right],$$
i.e., $f\in H_{\alpha+\rho}^p(\omega).$ This completes the proof of the sufficiency part in
(\ref{7}).

Let the right part of (\ref{8}) be true, and $f\in H^p_{\alpha+\rho}[\omega]$. First let us prove
that
\begin{eqnarray}\label{vallep2}
\left\| V_{n}^{(\alpha+r)} (f) \right\|_p \le C(\alpha,r) \frac{n^r}{\lambda_n} \left\|
V_{2n}^{(\alpha)} (f^{(\lambda,\beta)}) \right\|_p.
\end{eqnarray}
If $\beta=\rho+2m$, and therefore, $r=\rho$, then $V_{n}^{(\alpha+r)} (f) = V_{n}^{(\alpha)}
\Bigl(\{\frac{\nu^r}{\lambda_\nu}\}, f^{(\lambda,\beta)}\Bigr)$ and, by Lemma \ref{my2}
\begin{eqnarray*}
\left\| V_{n}^{(\alpha+r)} (f) \right\|_p \le C(\alpha,r) \frac{n^r}{\lambda_n}
 \left\| V_{n}^{(\alpha)} (f^{(\lambda,\beta)}) \right\|_p
\le C(\alpha,r) \frac{n^r}{\lambda_n}
  \left\| V_{2n}^{(\alpha)} (f^{(\lambda,\beta)}) \right\|_p.
\end{eqnarray*}
If $\beta\ne\rho+2m$, and therefore, $r>\rho$, then by Lemma \ref{my222}
\begin{eqnarray*}
\left\| V_{2^n}^{(\alpha+r)} (f) \right\|_p &\le& \sum\limits_{\nu=0}^{n} \left\|
V_{2^\nu}^{(\alpha+r)} (f) - V_{2^{\nu-1}}^{(\alpha+r)} (f)\right\|_p +
 \left\| V_{1}^{(\alpha+r)} (f) \right\|_p
\\
&\le& C
\sum\limits_{\nu=0}^{n} \frac{2^{\nu r}}{\lambda_{2^\nu}} \left\| V_{2^\nu}^{(\alpha)}
(f^{(\lambda,\beta)}) - V_{2^{\nu-1}}^{(\alpha)} (f^{(\lambda,\beta)})\right\|_p +
\frac{1}{\lambda_{1}}
 \left\| V_{1}^{(\alpha)} (f^{(\lambda,\beta)}) \right\|_p
\\
&\le& C
\left\| V_{2^{n+1}}^{(\alpha)} (f^{(\lambda,\beta)})\right\|_p
\left(\sum\limits_{\nu=0}^{n} \frac{2^{\nu r}}{\lambda_{2^\nu}} + \frac{1}{\lambda_{1}}\right)
 \le C(\alpha,r) \frac{2^{n r}}{\lambda_{2^n}}
  \left\| V_{2^{n+1}}^{(\alpha)} (f^{(\lambda,\beta)}) \right\|_p.
\end{eqnarray*}
 Thus, by (\ref{vallep2}) and the estimate
  $E_n(f)_p \le
\frac{C}{\lambda_n}E_{\left[\frac n4\right]}(f^{(\lambda,\beta)})_p,$ we can write
\begin{eqnarray*}
\omega_{\alpha+r}\left( f, \frac{1}{n}\right)_p
&\le& C(\alpha,r) \left(
{n}^{-(\alpha+r)}
\left\| V_{n}^{(\alpha+r)} (f) \right\|_p +
E_n(f)_p \right)\\
&\le&
\frac{C(\alpha,r)}{\lambda_n}
\left(
{n}^{-\alpha}
\left\| V_{n}^{(\alpha)} (f^{(\lambda,\beta)}) \right\|_p
  +   E_{\left[\frac n4\right]}(f^{(\lambda,\beta)})_p
\right) \\
&\le&
\frac{C(\alpha,r)}{\lambda_n}
\omega_{\alpha}\left( f^{(\lambda,\beta)}, \frac{1}{n}\right)_p
= O \left[
 \frac{\varphi\left(\frac{1}{n} \right)}{\lambda_{n}}\right]=
O \left[\omega\left(\frac{1}{n} \right)\right].
\end{eqnarray*}
Thus, $\frac{\varphi\left(\frac{1}{n} \right)}{\lambda_{n}} = O \left[\omega\left(\frac{1}{n}
\right)\right]$ is sufficient for $W_p^{\lambda,\beta}H_{\alpha}[\varphi]
 \subset H_{\alpha+r}^p[\omega]$.



\section{Proof of necessity in Theorem 1.}

First we define the trigonometric polynomials $\tau_{n+1} (x)$:
$$
\tau_{n+1} (x) = \sum\limits_{j=1}^{n+1}\alpha_j^{n} \sin j x, \qquad\mbox{where} \quad
\alpha_j^{n}=
\begin{cases} \frac{j}{n+2}, &\text{ $1\le j \le \frac{n+2}{2}$}
\\
1- \frac{j}{n+2}, &\text{ $ \frac{n+2}{2}\le j \le {n+1}$}.
\end{cases}
$$

We will use the following lemmas   as well as Lemmas \ref{my}-\ref{4l22}.

\begin{lemma} \label{lemmagejt} \textnormal{(\cite{tel})}.
Let $f(x)\in L_1$ have the Fourier series (\ref{fourier}). Then
$$E_n(f)_1 \ge C \left|
\sum\limits_{\nu=n+1}^\infty \frac{b_\nu}{\nu}\right|.$$
\end{lemma}


\begin{lemma} \textnormal{(\cite[ V. 1, p. 215]{zygm})}
\label{stlak} Let $1 \le p < \infty$.
\\
\textnormal{(a)}  If a function $f(x)\in L_p$ has the Fourier
series $\sum\limits_{\nu=1}^{\infty}(a_\nu\cos 2^\nu x+b_\nu\sin
2^\nu x)$, then
\begin{equation*}
\left\{\sum\limits_{\nu=1}^{\infty} \left( a^2_\nu + b^2_\nu\right)
\right\}^\frac{1}{2} \le C
\|f\|_p.
\end{equation*}
\textnormal{(b)}  Let $a_n, b_n (n\in \mathbf{N})$ be real
numbers such that $\sum\limits_{\nu=1}^{\infty} \left( a^2_\nu +
b^2_\nu\right)<\infty.$ Then
$\sum\limits_{\nu=1}^{\infty}(a_\nu\cos 2^\nu x +b_\nu\sin 2^\nu
x)$ be the Fourier series of a function $f(x)\in L_p$, and
\begin{equation*}
 \|f\|_p\le C
\left\{\sum\limits_{\nu=1}^{\infty} \left( a^2_\nu + b^2_\nu\right)
\right\}^\frac{1}{2}.
\end{equation*}
\end{lemma}


\begin{lemma} \textnormal{(\cite[Vol.2, p. 269]{baritext})}.\label{stlak1} Let $f(x)\in L_\infty$
have the Fourier series  $\sum\limits_{\nu=1}^{\infty} (a_\nu \cos 2^\nu x + b_\nu \sin 2^\nu x),$
where  $a_\nu, b_\nu \ge 0$. Then
$$C_1 \sum\limits_{\xi=n}^{\infty}
(a_\xi+b_\xi) \le  E_{2^n-1}(f)_\infty \le C_2 \sum\limits_{\xi=n}^{\infty} (a_\xi+b_\xi) .$$
\end{lemma}

We need the following definitions. Let $\omega(\cdot)\in\Phi_{\alpha}$.
\\
We call $\psi$ a {\em $Q_{\alpha,\, \theta}(\omega)$-sequence} if
\begin{eqnarray}
0&<& \psi_n \,\,\le\,\, n^{\alpha} \omega \left( \frac{1}{n}\right),\quad \psi_n \uparrow
\,(n\uparrow)\label{zv2}
\\
C_1 \,\omega \left( \frac{1}{n}\right) &\le& \left\{ \sum\limits_{\nu=n}^\infty
\nu^{-\alpha\theta-1}  \psi_\nu^\theta \right\}^\frac{1}{\theta} \le\, C_2 \,\omega \left(
\frac{1}{n}\right). \label{27}
\end{eqnarray}
We call $\varepsilon$ a {\em $q_{\alpha, \,\theta}(\omega)$-sequence} if
\begin{eqnarray}
0&<& \varepsilon_n \,\,\le\,\, \omega \left( \frac{1}{n+1}\right),\quad \varepsilon_n \downarrow
\,(n\uparrow)\label{32}
\\
C_1 \,\omega \left( \frac{1}{n+1}\right)
&\le&
 \left\{ (n+1)^{-\alpha\theta}
\sum\limits_{\nu=1}^{n+1} \nu^{\alpha\theta-1}  \varepsilon_\nu^\theta \right\}^\frac{1}{\theta}
\le\,
 C_2 \,\omega \left( \frac{1}{n+1}\right). \label{33}
\end{eqnarray}


\begin{center}{\bf Necessity in (\ref{1}) - (\ref{8}).}
\end{center}
We prove the necessity part by constructing corresponding examples. The proof is in eight steps.

\textbf{I.} $1<p<\infty$. {\sf {Step 1.} } Let us show the necessity part in (\ref{1}).
\\
Let $\omega(\cdot)\in\Phi_{\alpha+\rho}$ and $\theta=\min(2, p)$. We will construct a sequence
$\psi$ that will be a $Q_{\alpha+\rho,\, \theta}(\omega)$-sequence.
\\
Let us assume that we have chosen integers $1=n_1<n_2<\cdots<n_s$. Then, we define $n_{s+1}$ as
minimum of integers  $N>n_s$ such that
$$
 \omega \left( \frac{1}{N}\right)<
\frac{1}{2} \,\,\omega \left( \frac{1}{n_s}\right) \le  \omega \left( \frac{1}{N-1}\right).$$ We
set
$$
\psi_n= \begin{cases} n_s^{\rho+\alpha} \omega \left( \frac{1}{n_s}\right), &\text{if} \quad
n_s\le n <n_{s+1},
\\
0, &\text{if} \quad n=0.
\end{cases}$$
It is easy to see that this sequence is required.

Let $H^p_{\alpha+\rho}[\omega] \subset W_p^{ \lambda,\beta}$, and let the series in (\ref{1}) be
divergent. By means of properties of sequence $\left\{\psi_n\right\}$, we have
\begin{eqnarray*}
\infty=
\sum\limits_{\nu=1}^\infty
\left(
\lambda_{\nu+1}^\theta - \lambda_{\nu}^\theta \right)
\omega^\theta \left( \frac{1}{\nu}\right) &\le&
 C(\alpha,\rho,\theta)
\sum\limits_{\nu=1}^\infty \left( \lambda_{\nu+1}^\theta - \lambda_{\nu}^\theta \right)
\sum\limits_{m=\nu}^\infty m^{-(\alpha+\rho)\theta-1} \psi_m^\theta
\\
 &\le&
 C(\alpha,\rho,\theta)
\sum\limits_{\nu=1}^\infty
\lambda_{\nu}^\theta
\nu^{-(\alpha+\rho)\theta-1}
\psi_\nu^\theta.
\end{eqnarray*}
\\
{\sf Step 1(a): $2\le p <\infty$}.
  We consider the series
\begin{equation}\label{28}
\sum\limits_{\nu=1}^\infty 2^{-\nu(\alpha+\rho)} \left( \psi_{2^\nu}^2- \psi_{2^{\nu-1}}^2
\right)^\frac{1}{2} \cos 2^\nu x.
\end{equation}
Since
\begin{eqnarray}
\label{zv3} \sum\limits_{\nu=1}^\infty 2^{-2\nu(\alpha+\rho)} \left( \psi_{2^\nu}^2-
\psi_{2^{\nu-1}}^2 \right) &\le& \sum\limits_{\nu=1}^\infty
 \left(
\psi_{2^\nu}^2- \psi_{2^{\nu-1}}^2 \right) \sum\limits_{\xi=\nu}^\infty 2^{-2\xi(\alpha+\rho)}
\nonumber\\
&\le&
\sum\limits_{\nu=1}^\infty
2^{-2\nu(\alpha+\rho)}
\psi_{2^\nu}^2
\le  C \omega^2(1),
\end{eqnarray}
then, by Zygmund's Lemma \ref{stlak}, series (\ref{28}) is the
Fourier series of a function $f_1(x)\in L_p$. By Lemmas \ref{my}
and \ref{stlak},
$$
C(\alpha,\rho)
 \omega_{\alpha+\rho}\Bigl(f_1,\frac {1}{2^{n}}\Bigr)_p \le  2^{-n(\alpha+\rho)}
\left(
 \sum\limits_{\nu=1}^{n} a_\nu^2 2^{2(\alpha+\rho) \nu}\right)^\frac{1}{2}+
\left(\sum\limits_{\nu={n+1}}^{\infty} a_\nu^2 \right)^{\frac{1}{2}}=: I_1+I_2,$$ where
$a_\nu=2^{-(\alpha+\rho)}\left( \psi_{2^\nu}^2- \psi_{2^{\nu-1}}^2 \right)^\frac{1}{2}.$ Repeating
the argument  in (\ref{zv3}), we get
$$I_1\le
2^{-n(\alpha+\rho)} \psi_{2^n} \le
 \omega\Bigl(\frac{1}{2^n}\Bigr),$$
because of (\ref{zv2}) and
$$I_2\le C(\alpha,\rho)
\left( \sum\limits_{\nu=n+1}^\infty  2^{-2\nu(\alpha+\rho)} \psi_{2^\nu}^2 \right)^\frac{1}{2} \le
C(\alpha,\rho) \omega\Bigl(\frac{1}{2^n}\Bigr),$$ because of  (\ref{27}).

Thus, $f_1(x)\in H^p_{\alpha+\rho}[\omega].$ Then from our
assumption, $f_1(x)\in W_p^{\lambda,\beta}$. On the other hand,
$$\left\|f_1^{(\lambda,\beta)}\right\|_p \ge
 C(\alpha,\rho,\theta)
\left( \sum\limits_{\nu=1}^\infty \lambda_{\nu}^2
\nu^{-2(\alpha+\rho)-1} \psi_\nu^2 \right)^\frac{1}{2}\, =
\,\infty.$$ This contradiction proves the convergence of series
in (\ref{1}).
\\
{\sf Step 1(b): $1<p\le 2$}.
  Consider series\footnote{Series of this type was considered in \cite{mtiman}.}
\begin{equation}\label{29}
\psi_{1} \cos  x +
\sum\limits_{\nu=1}^\infty
2^{-\nu(\alpha+\rho)}
2^{\nu(\frac{1}{p}-1)}
\left(
\psi_{2^\nu}^p- \psi_{2^{\nu-1}}^p
\right)^\frac{1}{p}
\sum\limits_{\mu=2^{\nu-1}+1}^{2^{\nu}}
\cos \mu x.
\end{equation}
Using Jensen inequality\,
 $\left( \sum\limits_{n=1}^{\infty} a_n^\alpha \right)^{1/\alpha} \le
\left( \sum\limits_{n=1}^{\infty} a_n^\beta \right)^{1/\beta}$ ($a_n\ge 0$ and $\,0 < \beta \le
\alpha<\infty$), we write
\begin{eqnarray*}
& &
\int\limits_0^{2\pi}
\left[
\sum\limits_{\nu=1}^\infty
\left(
2^{-\nu(\alpha+\rho)}
2^{\nu(\frac{1}{p}-1)}
\left(
\psi_{2^\nu}^p- \psi_{2^{\nu-1}}^p
\right)^\frac{1}{p}
\sum\limits_{\mu=2^{\nu-1}+1}^{2^{\nu}}
\cos \mu x
\right)^2
\right]^\frac{p}{2} dx
\\
&\le& \int\limits_0^{2\pi} \left[ \sum\limits_{\nu=1}^\infty 2^{-\nu p(\alpha+\rho)} 2^{\nu(1-p)}
\left( \psi_{2^\nu}^p- \psi_{2^{\nu-1}}^p \right) \left| \sum\limits_{\mu=2^{\nu-1}+1}^{2^{\nu}}
\cos \mu x \right|^p \,\right] dx
\\
&\le& C(p) \sum\limits_{\nu=1}^\infty \left( \psi_{2^\nu}^p- \psi_{2^{\nu-1}}^p \right) 2^{-\nu
p(\alpha+\rho)} \le C(p) \omega^p(1),
\end{eqnarray*}
because of $\,\,C_1(p) 2^{\nu(p-1)}\le  \left\| \sum\limits_{\mu=2^{\nu-1}+1}^{2^{\nu}} \cos \mu x
\right\|^p_p \le C_2(p)2^{\nu(p-1)}$.
 By the Littlewood-Paley theorem, there exists a function $f_2\in L_p$ with Fourier
series (\ref{29}). One can see that $f_2 \in H^p_{\alpha+\rho}[\omega].$ Then $f_2\in
W_p^{\lambda,\beta}$. On the other hand, Paley's theorem on Fourier coefficients \cite[V.2, p.
121]{zygm} implies that for $f_2\in L_p$
\begin{eqnarray*}
\left\|f_2^{(\lambda,\beta)}\right\|_p^p &\ge&
 C(p)
\sum\limits_{\nu=1}^\infty
2^{-\nu p(\alpha+\rho)}
2^{\nu(1-p)}
\left(
\psi_{2^\nu}^p- \psi_{2^{\nu-1}}^p
\right)
\sum\limits_{\mu=2^{\nu-1}+1}^{2^{\nu}}
\lambda_\mu^p \mu^{p-2}
\\
&\ge&
 C(\alpha,\rho, p)
\sum\limits_{\nu=1}^\infty
\left(
\psi_{2^\nu}^p- \psi_{2^{\nu-1}}^p
\right)
\sum\limits_{\xi=\nu}^\infty
\left(
2^{-\xi(\alpha+\rho)p}\lambda_{2^\xi}^p
-
2^{-(\xi+1)(\alpha+\rho)p}\lambda_{2^{\xi+1}}^p
\right)
\\
&\ge&
 C_1(\alpha,\rho,p)
\sum\limits_{\nu=1}^\infty
\psi_\nu^p \lambda_{\nu}^p
\nu^{-p(\alpha+\rho)-1}
-
 C_2(\alpha,\rho,p)
 \psi_1^p \lambda_{2}^p
2^{-p(\alpha+\rho)-1}\, =
\,\infty.
\end{eqnarray*}
This contradiction shows that the series in the right part of (\ref{1}) converges. This completes
the proof of the necessity part of (\ref{1}).

{\sf {Step 2.} } Let us prove the necessity in (\ref{2}) for $2\le p <\infty.$
\\
We notice that by Lemmas \ref{my} and \ref{stlak}, we have for $f(x)\sim
\sum\limits_{\nu=1}^{\infty} (a_\nu \cos 2^\nu x + b_\nu \sin 2^\nu x)$
\begin{equation}\label{mod-cF}
\omega_\alpha\left(f, \frac{1}{2^m}\right)_p \asymp \left( 2^{-2m\alpha} \sum\limits_{\nu=1}^{m}
(a_\nu^2 + b_\nu^2) 2^{2\nu\alpha}\right)^\frac{1}{2}+ \left(\sum\limits_{\nu=m+1}^{\infty}
(a_\nu^2 + b_\nu^2)\right)^\frac{1}{2}.
\end{equation}
Let $\omega(\cdot)\in\Phi_{\alpha+\rho}$. One can construct\footnote{See, for example,
\cite{gejt}.} a sequence $\varepsilon$ such that $\varepsilon$ is a $q_{\alpha+\rho,
\,\theta}(\omega)$-sequence. We consider for this case
\begin{equation}\label{34}
\varepsilon_0+ \left( \varepsilon_{1}^2 - \varepsilon_{2}^2\right)^\frac{1}{2} \cos  x +
\sum\limits_{\nu=1}^\infty \left( \varepsilon_{2^{\nu}}^2 -
\varepsilon_{2^{\nu+1}}^2\right)^\frac{1}{2} \cos 2^{\nu} x.
\end{equation}
Repeating the argument for series (\ref{28}) we obtain that series (\ref{34}) is the Fourier
series of a function $f_3\in L_p$. Since $E_{2^{n-1}}(f_3)_p\le C(p)\varepsilon_{2^{n}}$, then by
(\ref{a3}), (\ref{33}) implies $f_3\in H_{\alpha+\rho}^p[\omega]$. We define
$f_{13}:=f_{1}+f_{3}.$ We have $f_{13}\in H_{\alpha+\rho}^p[\omega]\subset W_p^{ \lambda,\beta}
H_\alpha[\varphi]$.
 It is easy to see from (\ref{mod-cF}) that
 \begin{equation}\label{99}
C(\alpha, \beta) \, \omega_{\alpha}\Bigl(f_{13}^{(\lambda,\beta)}, \frac{1}{n+1}\Bigr)_p \ge
\omega_{\alpha}\Bigl(f_{1}^{(\lambda,\beta)}, \frac{1}{n+1}\Bigr)_p+
\omega_{\alpha}\Bigl(f_{3}^{(\lambda,\beta)}, \frac{1}{n+1}\Bigr)_p.
 \end{equation}

Let us estimate $\omega_{\alpha}\Bigl(f_{1}^{(\lambda,\beta)}, \frac{1}{n+1}\Bigr)_p.$
 By (\ref{mod-cF}),
using the properties of the sequence  $\{\psi_{\nu}\}$,
 we have ($2^m\le n+1< 2^{m+1}$)
\begin{eqnarray}
\omega^2_{\alpha}\left(f_1^{(\lambda,\beta)}, \frac{1}{n+1}\right)_p &\ge&  C(\alpha, \beta)
\sum\limits_{\nu=m}^\infty \psi_{2^\nu}^2 2^{-2\nu(r+\alpha)} \left(
 \sum\limits_{k=m}^{\nu} (\lambda_{2^k}^2 - \lambda_{2^{k-1}}^2) +
 \lambda_{2^{m-1}}^2\right)
 \nonumber
\\
&\ge& C(\alpha, \beta) \left( \lambda_{2^m}^2 \omega^2\left(\frac{1}{2^m}\right) +
\sum\limits_{k=m}^\infty (\lambda_{2^k}^2 - \lambda_{2^{k-1}}^2)
\omega^2\left(\frac{1}{2^k}\right)\right)
 \nonumber
 \\
&\ge& C(\alpha, \beta) \left(
 \lambda_{n+1}^2 \omega^2\left(\frac{1}{n+1}\right) + \sum\limits_{\nu={n+2}}^\infty
\left(\lambda_{\nu+1}^2 -\lambda_{\nu}^2\right) \omega^2\left(\frac{1}{\nu}\right) \right) .
 \label{f1}
\end{eqnarray}

Now we estimate $\omega_{\alpha}\Bigl(f_{3}^{(\lambda,\beta)}, \frac{1}{n+1}\Bigr)_p.$ By
(\ref{mod-cF}), we have ($2^m\le n+1< 2^{m+1}$):
\begin{eqnarray} \label{35}
\omega_\alpha^2\left(f_3^{(\lambda,\beta)}, \frac{1}{n+1}\right)_p &\ge& C(\alpha, \beta) 2^{-2 m
\alpha}
 \sum\limits_{\nu=0}^m 2^{2 \nu \alpha} \lambda_{2 \nu}^2
\left( \varepsilon_{2^{\nu}}^2 - \varepsilon_{2^{\nu+1}}^2\right) \nonumber
\\
&\ge& C_1(\alpha, \beta) 2^{-2 m \alpha}
 \sum\limits_{\nu=0}^m 2^{2 \nu \alpha} \lambda_{2^\nu}^2
\varepsilon_{2^{\nu}}^2 - C_2(\alpha,p)    \lambda_{2^{m+1}}^2 \varepsilon_{2^{m+1}}^2.
\end{eqnarray}
The Jackson inequality implies
\begin{eqnarray} \label{36}
\omega_\alpha^2\left(f_3^{(\lambda,\beta)}, \frac{1}{n+1}\right)_p &\ge& C(\alpha)
E_{2^m-1}^2\left(f_3^{(\lambda,\beta)} \right)_p \ge C(\alpha, \beta)
 \sum\limits_{\nu=m}^\infty  \lambda_{2 \nu}^2
\left( \varepsilon_{2^{\nu}}^2 - \varepsilon_{2^{\nu+1}}^2\right) \nonumber \\
&\ge& C(\alpha, \beta)
 \lambda_{2^m}^2 \varepsilon_{2^{m}}^2.
\end{eqnarray}
Both estimates (\ref{35}) and (\ref{36}) imply
\begin{equation} \label{37}
\omega_\alpha\left(f_3^{(\lambda,\beta)}, \frac{1}{n+1}\right)_p \ge C(\alpha, \beta)
\left((n+1)^{-2\alpha}
 \sum\limits_{\nu=1}^{n+1}
  \lambda_{\nu}^2
\nu^{2\alpha-1} \varepsilon^2_{\nu}\right)^\frac{1}{2}.
\end{equation}
Using (\ref{33}) and $\nu^{-\rho}\lambda_\nu \downarrow$, we get
\begin{eqnarray}
  \lambda_{n+1}^2 \omega^2 \left( \frac{1}{n+1}\right)
&+& (n+1)^{-2\alpha} \sum\limits_{\nu=1}^{n+1} \nu^{2(\rho+\alpha)} \omega^2 \left(
\frac{1}{\nu}\right) \left(
  \lambda_{\nu}^2 \nu^{-2\rho}-
  \lambda_{\nu+1}^2 (\nu+1)^{-2\rho}
\right)\nonumber
\\
&\le& C(\alpha, \rho) (n+1)^{-2\alpha} \sum\limits_{\nu=1}^{n+1} \nu^{2\alpha-1}
 \lambda_{\nu}^2 \varepsilon_{\nu}^2.
\label{f3}
\end{eqnarray}
Combining estimates (\ref{99}), (\ref{f1}), (\ref{37}), (\ref{f3}), and
$\omega_{\alpha}\Bigl(f_{13}^{(\lambda,\beta)}, \frac{1}{n+1}\Bigr)_p =
O\left[\varphi\left(\frac{1}{n}\right)\right]$, we arrive at the condition in the right part of
(\ref{2}).

{\sf {Step 3.} } Let us prove the necessity in (\ref{2}) for $1<p<2.$ The proof for $1< p \le 2$
is similar to $2\le p <\infty$. The only difference is that we use Paley's theorem on Fourier
coefficients instead of Zygmund's theorem. In this case we consider the sum of $f_{2}(x)$ and the
following function
\begin{equation}\label{38}
\varepsilon_0+ \left( \varepsilon_{1}^p - \varepsilon_{2}^p\right)^\frac{1}{p} \cos  x +
\sum\limits_{\nu=0}^\infty 2^{\nu\left(\frac{1}{p}-1\right)} \left( \varepsilon_{2^{\nu+1}}^p -
\varepsilon_{2^{\nu+2}}^p\right)^\frac{1}{p} \sum\limits_{\mu=2^{\nu}+1}^{2^{\nu+1}} \cos \mu x.
\end{equation}

{\sf {Step 4.} } To prove the necessity in (\ref{3}) and (\ref{7}), we consider the general case
$1\le p\le \infty.$ Let $\Phi$ be the class of all decreasing null-sequences. It is clear that
\begin{equation*}\label{conditions}
\frac{1}{\lambda_n}= O \left[ \omega\left(\frac{1}{n}\right) \right] \qquad \Longleftrightarrow
\qquad \forall\,\,\gamma=\{\gamma_n\}\in \Phi\qquad \frac{\gamma_n}{\lambda_n}= O \left[
\omega\left(\frac{1}{n}\right) \right].
\end{equation*}
Let us assume that $\frac{\gamma_n}{\lambda_n}= O \left[\omega\left(\frac{1}{n}\right)\right]$
does not hold for all $\,\gamma\in \Phi$ and  $W^{\lambda,\beta}_p \subset
H^p_{\alpha+\rho}[\omega]$.
 Then there exist $\gamma=\{\gamma_n\}\in \Phi$ and
$\{C_{n}\uparrow \infty\}$ such that $\frac{\gamma_{m_n}}{\lambda_{m_n}}\ge C_{n}
\omega\left(\frac{1}{m_n}\right).$ Further, we choose a subsequence $\left\{m_{n_k} \right\}$ such
that $\frac{m_{n_{k+1}}}{m_{n_k}}\ge 2$ and $\gamma_{m_{n_k}}\le 2^{-k}.$ Consider the series
\begin{equation}\label{19vs}
\sum\limits_{k=0}^\infty \frac{\gamma_{m_{n_k}}}{\lambda_{m_{n_k}}} \cos ( {m_{n_k}}+1)x.
\end{equation}
Since $\sum\limits_{k=0}^\infty \frac{\gamma_{m_{n_k}}}{\lambda_{m_{n_k}}} \le
\frac{1}{\lambda_{m_{n_0}}} \sum\limits_{k=0}^\infty \frac{1}{2^k}<\infty,$ there exists a
function $f_4\in L_p$ with Fourier series (\ref{19vs}). Because
 $\sum\limits_{k=0}^\infty {\gamma_{m_{n_k}}} \le
\sum\limits_{k=0}^\infty \frac{1}{2^k}<\infty,$ we have $f_4^{(\lambda,\beta)}\in L_p$, i.e.,
$f_4\in W_p^{\lambda,\beta}$.

On the other hand, by (\ref{11}) and $E_{n-1}(f)_p \ge C \left(|a_n|+|b_n|\right)$,
\begin{eqnarray*}
\omega_{\alpha+\rho} \left(f_4, \frac{1}{m_{n_k}} \right)_p\ge C(\alpha, \rho)
E_{m_{n_k}}(f_{4})_p \ge C(\alpha, \rho) \frac{\gamma_{m_{n_k}}}{\lambda_{m_{n_k}}} \ge C(\alpha,
\rho) C_{n_k} \omega\left(\frac{1}{m_{n_k}}\right),
\end{eqnarray*}
i.e., $f_4\notin H^p_{\alpha+\rho}[\omega].$ This contradiction implies that the condition
$\frac{1}{\lambda_n}= O \left[\omega\left(\frac{1}{n}\right)\right]$ is necessary for
$W_p^{\lambda,\beta} \subset H^p_{\alpha+\rho}[\omega]$.


{\sf {Step 5.} } To prove the necessity in (\ref{4}) and (\ref{8}), we verify that for any
$\rho>0$ and for any $1\le p\le \infty$,
\begin{equation}\label{emb1}
W_p^{ \lambda,\beta} H_\alpha[\varphi]
 \subset H_{\alpha+\rho}^p[\omega]
\quad \Longrightarrow \quad
\frac{\varphi\left(\frac{1}{n} \right)}{\lambda_{n}}= O
\left[\omega\left(\frac{1}{n} \right)\right].
\end{equation}
First we remark that
\begin{equation*}
\frac{\varphi\left(\frac{1}{n} \right)}{\lambda_{n}}= O \left[\omega\left(\frac{1}{n}
\right)\right] \qquad \Longleftrightarrow \qquad \forall\,\,\gamma=\{\gamma_n\}\in \Phi\qquad
\frac{\gamma_n \varphi\left(\frac{1}{n} \right)}{\lambda_{n}}= O \left[\omega\left(\frac{1}{n}
\right)\right].
\end{equation*}
 Assume that the relation in the right hand side of (\ref{emb1}) does not hold.
Then there exist $\gamma=\{\gamma_n\}\in \Phi$ and $\{C_{n}\uparrow \infty\}$ such that
$\frac{\gamma_{m_n}\varphi\left(\frac{1}{m_n}\right)}{\lambda_{m_n}}\ge C_{n}
\omega\left(\frac{1}{m_n}\right).$ We choose a subsequence $\left\{m_{n_k} \right\}$ such that
$\frac{m_{n_{k+1}}}{m_{n_k}}\ge 2$ and $\gamma_{m_{n_k}}\le 2^{-k}.$ Because
$\sum\limits_{k=0}^\infty \gamma_{m_{n_k}}\varphi\left(\frac{1}{m_n}\right)
 \le
\varphi\left(\frac{1}{m_0}\right) \sum\limits_{k=0}^\infty \frac{1}{2^k}<\infty,$ there exists a
function $f_5\in L_p$ with Fourier series
\begin{equation}\label{19vsvs}
\sum\limits_{k=0}^\infty {\gamma_{m_{n_k}}}\varphi\left(\frac{1}{m_{n_k}}\right) \cos (
{m_{n_k}}+1)x.
\end{equation}
For $m_{n_k}\le n < m_{n_{k+1}}$, by Lemmas \ref{my} and
\ref{stlak1}, we have
\begin{eqnarray*}
\omega_{\alpha} \left(f_5, \frac{1}{n} \right)_p
 &\le&
 C \omega_{\alpha} \left(f_5, \frac{1}{n} \right)_\infty
 \le C \left( n^{-\alpha}
 \sum \limits_{s=0}^k
{\gamma_{m_{n_s}}}\varphi\left(\frac{1}{m_{n_s}}\right)
{m^{\alpha}_{n_s}} + \sum
\limits_{s=k+1}^\infty {\gamma_{m_{n_s}}}\varphi\left(\frac{1}{m_{n_s}}\right) \right) \\
 &\le&
C\left( \varphi\left(\frac{1}{n}\right)
 \sum \limits_{s=0}^k
{\gamma_{m_{n_s}}} + \varphi\left(\frac{1}{n}\right)
 \sum \limits_{s=k+1}^\infty
{\gamma_{m_{n_s}}} \right) \le  C \varphi\left(\frac{1}{n}\right).
\end{eqnarray*}
Therefore, $f_5\in H^p_{\alpha}[\varphi]$, i.e., setting $\frac{1}{\lambda} := \left\{
\frac{1}{\lambda_1}, \frac{1}{\lambda_2}, \frac{1}{\lambda_3}, \cdots \right\}$, we have
$f_5^{(\frac{1}{\lambda}, - \beta)} \in W_p^{ \lambda,\beta} H_\alpha[\varphi].$

However, we have
\begin{eqnarray*}
\omega_{\alpha+\rho} \left(f_5^{(\frac{1}{\lambda}, - \beta)}, \frac{1}{m_{n_k}} \right)_p\ge C
E_{m_{n_k}}(f_5^{(\frac{1}{\lambda}, - \beta)})_p \ge C \frac{\gamma_{m_{n_k}}
\varphi\left(\frac{1}{m_{n_k}}\right)}
{\lambda_{m_{n_k}}} \ge C C_{n_k}
\omega\left(\frac{1}{m_{n_k}}\right),
\end{eqnarray*}
i.e., $f_5^{(\frac{1}{\lambda}, - \beta)}\notin
H^p_{\alpha+\rho}[\omega].$ This contradicts our assumption.
 The proof of the necessity part in (\ref{3})-(\ref{4}) and (\ref{7})-(\ref{8}) is now complete.

\textbf{II.} $p=1$ or $p=\infty$. {\sf {Step 6.} } Let us prove the necessity in (\ref{5}). Let
$H_{\alpha+\rho}^p[\omega] \subset W_p^{\lambda,\beta}$ and the series in (\ref{5}) be divergent.


{\sf Step 6(a): $\sin \frac{\beta\pi}{2} \ne 0$}. Then the divergence of the series in (\ref{5})
is equivalent to the divergence of the series $\sum\limits_{n=1}^\infty \frac{\lambda_{n}}{n}
\omega\left(\frac{1}{n}\right).$

Let $p=1$. We take a sequence  $\varepsilon$ which is a $q_{\alpha+\rho, \,1}(\omega)$-sequence
and consider the series
\begin{equation}\label{39}
\sum\limits_{\nu=1}^\infty
\left(\varepsilon_\nu - \varepsilon_{\nu+1}\right) K_\nu(x).
\end{equation}
This series is convergent in $L_1$ (see \cite{gejt}) to a function $f_6(x)$,
$E_n(f_6)_1=O\left(\varepsilon_n\right)$. By means of  (\ref{a3}), (\ref{33}), we get $f_6 \in
H_{\alpha+\rho}^p[\omega] \subset W_p^{\lambda, \beta}$. One can rewrite (\ref{39}) in the
following form
$$
\sum\limits_{\nu=1}^\infty  a_\nu \cos \nu x, \qquad \mbox{where}  \qquad
 a_\nu=\varepsilon_\nu -\nu \sum\limits_{j=\nu}^\infty
\frac{\varepsilon_j - \varepsilon_{j+1}}{j+1}.
$$
By Lemma \ref{lemmagejt},
\begin{eqnarray*}
\left\| f_{6}^{(\lambda,\beta)} \right\|_1 &\ge& C(\beta) \sum\limits_{\nu=1}^\infty
\frac{\lambda_\nu}{\nu} a_{\nu} =
 C(\beta)
 \left(
\sum\limits_{\nu=1}^\infty \frac{\lambda_\nu}{\nu}\varepsilon_{\nu} - \sum\limits_{\nu=1}^\infty
\lambda_\nu \sum\limits_{j=\nu}^\infty \frac{\varepsilon_{j} - \varepsilon_{j+1}}{j+1} \right)
\\
&=& C(\beta) \left( \sum\limits_{\nu=1}^\infty \frac{\lambda_\nu}{\nu}\varepsilon_{\nu} -
\sum\limits_{\nu=1}^\infty (a_{\nu}-a_{\nu+1}) \lambda_{\nu}
 \right)
\\
&\ge& C_1(\beta)
 \sum\limits_{\nu=1}^\infty
\frac{\lambda_\nu}{\nu}\varepsilon_{\nu} - C_2(\beta) \left(
 \lambda_{1} a_{1} +
 \sum\limits_{n=1}^\infty
 (\lambda_{n+1}-\lambda_{n})a_{n} \right).
 \end{eqnarray*}
Using (\ref{32}) and (\ref{33}), we get
\begin{eqnarray*}
 \sum\limits_{\nu=1}^\infty
\frac{\lambda_\nu}{\nu}\varepsilon_{\nu} &\le& C(\rho)
 \sum\limits_{\nu=1}^\infty
\frac{\lambda_\nu}{\nu} \omega\left(\frac{1}{\nu}\right)  \le C(\rho)
 \sum\limits_{\nu=1}^\infty
{\lambda_\nu} \nu^{-(\alpha+\rho)-1}
 \sum\limits_{m=1}^\nu
m^{\alpha+\rho-1} \varepsilon_{m}
\\
&=& C(\rho)
 \sum\limits_{m=1}^\infty m^{\alpha+\rho-1} \varepsilon_{m}
 \sum\limits_{\nu=m}^\infty
\frac{\lambda_\nu}{\nu^\rho} \nu^{-\alpha-1} \le C(\rho, \alpha)
 \sum\limits_{m=1}^\infty
\frac{\lambda_m}{m}\varepsilon_{m}.
 \end{eqnarray*}
Therefore,
\begin{equation}\label{41}
\left\| f_6^{(\lambda,\beta)} \right\|_1 \ge C_1(\alpha, \rho, \beta)
 \sum\limits_{\nu=1}^\infty \lambda_\nu \nu^{-1}
\omega\left(\frac{1}{\nu}\right)- C_2(\alpha, \rho, \beta) \left( \lambda_1 a_1 +
\sum\limits_{\nu=1}^\infty (\lambda_{\nu+1} - \lambda_{\nu}) a_\nu \right).
\end{equation}
On the other hand, using monotonicity of $\{a_\nu\}$ and Lemma \ref{lemmagejt}, we have
\begin{eqnarray}\label{42}
C(\beta, \rho)\left\| f_6^{(\lambda,\beta)} \right\|_1 &\ge& C(\rho) \left( \lambda_1 a_1 +
\sum\limits_{\nu=1}^\infty \frac{\lambda_{\nu} a_\nu}{\nu} \right) \ge C(\rho) \left( \lambda_1
a_1 + \sum\limits_{\nu=0}^\infty \lambda_{2^\nu} a_{2^\nu} \right)
\nonumber\\
&\ge& \lambda_1 a_1 + \sum\limits_{\nu=0}^\infty  a_{2^\nu} ( \lambda_{2^{\nu+1}} -
\lambda_{2^{\nu}}) \ge \lambda_1 a_1 + \sum\limits_{\nu=1}^\infty (\lambda_{\nu+1} -
\lambda_{\nu}) a_\nu.
\end{eqnarray}
From  (\ref{41}) and (\ref{42}) we get
$$
\left\| f_6^{(\lambda,\beta)} \right\|_1 \ge C(\alpha, \rho, \beta) \sum\limits_{\nu=1}^\infty
\lambda_\nu \nu^{-1} \omega\left(\frac{1}{\nu}\right) = \infty. $$ This contradiction implies the
convergence of series in (\ref{5}).

Let now $p=\infty$.  Define the function (see \cite{bari-sopr}) $f_7(x) =
\sum\limits_{\nu=1}^\infty \varepsilon_\nu \nu^{-1} \sin \nu x,$ where $\varepsilon$ is a
$q_{\alpha+\rho, \,1}(\omega)$-sequence.
 We have $E_n(f_7)_\infty \le
C\varepsilon_{n+1}$. Using (\ref{a3}) and (\ref{33}), we get $f_7\in
H_{\alpha+\rho}^p[\omega]\subset W_p^{\lambda,\beta}$. On the other hand,
\begin{eqnarray*}
\infty = \left\| f_7^{(\lambda,\beta)} \right\|_\infty &\ge& C(\beta)
 \sum\limits_{\nu=0}^\infty 2^{\nu(\alpha+\rho)}
\varepsilon_{2^\nu} \frac{\lambda_{2^\nu}}{2^{\nu(\alpha+\rho)}}
\\
&\ge& C(\alpha, \rho, \beta) \sum\limits_{\nu=0}^\infty 2^{\nu(\alpha+\rho)}\varepsilon_{2^\nu}
\sum\limits_{m=\nu}^\infty \frac{\lambda_{2^m}}{2^{m(\alpha+\rho)}}
\\
&\ge&  C(\alpha, \rho, \beta)
 \sum\limits_{\nu=1}^\infty
\lambda_\nu \nu^{-1} \omega\left(\frac{1}{\nu}\right).
\end{eqnarray*}
This implies  the convergence of  the series in (\ref{5}).


{\sf Step 6(b): $\sin \frac{\beta\pi}{2} = 0$}. Let the series in (\ref{5}) be divergent. We will
consider only the non-trivial case $\rho>0.$ Let $\varepsilon$ be a $q_{\alpha+\rho,
\,1}(\omega)$-sequence. By means of the properties  $\{\lambda_n\}$, we have
\begin{eqnarray}
\sum\limits_{\nu=2}^{\infty}
\left(
  \lambda_{\nu+1} - \lambda_{\nu}
\right) \omega \left( \frac{1}{\nu}\right)
&\le&
C(\alpha, \rho)
\sum\limits_{s=0}^{\infty}
\left(\varepsilon_{2^s} - \varepsilon_{2^{s+1}} \right)
\left[
\sum\limits_{m=0}^{s}
\right.
2^{m(\alpha+\rho)}
\sum\limits_{\nu=m}^{s}
2^{-\nu(\alpha+\rho)} \left(
  \lambda_{2^\nu} - \lambda_{2^{\nu-1}}  \right)
\nonumber
\\
&+&
\left.\sum\limits_{m=0}^{s}
2^{m(\alpha+\rho)}
\sum\limits_{\nu=s+1}^{\infty}
2^{-\nu(\alpha+\rho)}
\left(
  \lambda_{2^\nu} - \lambda_{2^{\nu-1}}  \right)
 \right]
\nonumber
\\
&\le&
C(\alpha, \rho)
\sum\limits_{\nu=0}^{\infty}
\left(\varepsilon_{\nu} - \varepsilon_{\nu+1} \right)
  \lambda_{\nu}.
\label{43}
\end{eqnarray}
Let $p=1$.  The series
\begin{equation}\label{44}
\sum\limits_{\nu=1}^\infty \left(\varepsilon_{\nu} - \varepsilon_{\nu+1} \right) \tau_\nu(x)
\end{equation}
converges (\cite{gejt}) to a $f_8\in L_1$, and $E_n(f_8)_1\le C \varepsilon_{n+1}$. Then $f_8\in
H_{\alpha+\rho}^p[\omega]\subset W_p^{\lambda,\beta}$. We rewrite (\ref{44}) in the following way
$$
\sum\limits_{\nu=1}^\infty  b_\nu \sin \nu x, \qquad \mbox{where}  \qquad
 b_\nu=
\sum\limits_{j=\nu}^{2\nu-2}
 \left( 1 - \frac{\nu}{j+1} \right)
\left(\varepsilon_{\nu} - \varepsilon_{\nu+1} \right)
+ \sum\limits_{j=2\nu-1}^\infty
 \frac{\nu}{j+1}
\left(\varepsilon_{\nu} - \varepsilon_{\nu+1} \right).
$$
By Lemma \ref{lemmagejt}, we write
\begin{equation*}
\left\| f_8^{(\lambda,\beta)} \right\|_1 \ge C(\beta) \sum\limits_{\nu=1}^\infty \lambda_\nu
\frac{b_\nu}{\nu}  \ge C(\beta) \sum\limits_{\nu=0}^\infty \lambda_\nu \left(\varepsilon_{\nu} -
\varepsilon_{\nu+1} \right).
\end{equation*}
This contradicts the divergence of the series in (\ref{43}). Thus, the series in (\ref{5})
converges.

Let $p=\infty$. Let
 $\psi$ be a $Q_{\alpha+\rho,1}$-sequence. Then, we define
\begin{equation*}
f_9(x) = \psi_1 \cos x+ \sum\limits_{\nu=1}^\infty 2^{-\nu(\rho+\alpha)} \left( \psi_{2^\nu} -
\psi_{2^{\nu-1}} \right) \cos  2^\nu x.
\end{equation*}
Similarly, as for $f_1$ in the case $2\le p <\infty,$ it is easy to see that $f_9\in
H_{\alpha+\rho}^p[\omega]\subset W_p^{\lambda,\beta}$. By Lemma \ref{stlak1}, we have
\begin{eqnarray*}
\left\| f_9^{(\lambda,\beta)} \right\|_\infty &\ge& C(\beta) \left(\lambda_1 \psi_1+
\sum\limits_{\nu=1}^\infty \lambda_{2^\nu} 2^{-\nu(\rho+\alpha)} \left(\psi_{2^\nu}
-\psi_{2^{\nu-1}} \right) \right)
\\
& \ge& C(\beta) \sum\limits_{\nu=1}^\infty \lambda_{\nu} \nu^{-(\rho+\alpha)-1} \psi_{\nu}
\\
& \ge& C(\beta) \sum\limits_{\nu=1}^\infty \left(\lambda_{\nu+1} - \lambda_{\nu} \right)
\omega(1/\nu) =\infty,
\end{eqnarray*}
this contradicts $f_9\in W_p^{\lambda,\beta}$. This completes the proof of the necessity in
(\ref{5}).

{\sf {Step 7.} } We will prove the necessity in (\ref{6}) for the case $\sin \frac{\pi\beta}{2}\ne
0$. Let $H_{\alpha+r}^p[\omega] \subset W_p^{\lambda,\beta}H_\alpha[\varphi]$. Let $\varepsilon$
be a $q_{\alpha+r, \,1}(\omega)$-sequence. So, (\ref{33}) holds for $\alpha+r$ instead of $\alpha$
and $1$ instead of $\theta$. Since $\sin \frac{\pi\beta}{2}\ne 0$,
\begin{eqnarray}\label{45}
J&:=& \lambda_{n+1} \omega\left(\frac{1}{n+1}\right) + n^{-\alpha} \sum\limits_{\nu=1}^n
\nu^{r+\alpha} \left(\nu^{-r}\lambda_{\nu}- (\nu+1)^{-r}\lambda_{\nu+1} \right)
\omega\left(\frac{1}{\nu}\right) \nonumber
\\
&+& | \cos \frac{\beta\pi}{2} | \sum\limits_{\nu=n+2}^\infty
\left(\lambda_{\nu+1}-\lambda_{\nu}\right) \omega\left(\frac{1}{\nu}\right) + | \sin
\frac{\beta\pi}{2} | \sum\limits_{\nu=n+2}^\infty
 \lambda_\nu
\frac{\omega\left(\frac{1}{\nu}\right)}{\nu} \nonumber
\\
&\le& C(\alpha, \beta, r) \left( \sum\limits_{\nu=n+1}^\infty
 \lambda_\nu \frac{\varepsilon_\nu}{\nu}
+ n^{-\alpha} \sum\limits_{\nu=1}^n \lambda_\nu {\varepsilon_\nu} \nu^{\alpha-1} \right) =:
C(\alpha, \beta, r) \left(J_1+J_2\right).
\end{eqnarray}
We  will use several times the following evident inequalities:
\begin{equation}\label{f+-}
\omega_\alpha\Bigl(f, \frac{1}{n}\Bigr)_p \asymp \omega_\alpha\Bigl(f_{+}, \frac{1}{n}\Bigr)_p +
\omega_\alpha\Bigl(f_{-}, \frac{1}{n}\Bigr)_p, \qquad\mbox{where}\quad f_{\pm}(x):= \frac{f(x) \pm
f(-x)}{2}.
\end{equation}
\\
{\sf Step 7(a): $p=\infty$ and $\cos \frac{\pi\alpha}{2}\ne 0$}.
 Then by Lemma \ref{my}, we write $\Bigl($ $f_{7+}^{(\lambda,\beta)}:=\bigl(f_{7}^{(\lambda,\beta)}\bigr)_+$
$\Bigr)$
\begin{eqnarray}\label{vspom78}
\omega_\alpha\Bigl(f_{7}^{(\lambda,\beta)}, \frac{1}{n}\Bigr)_p &\ge&
\omega_\alpha\Bigl(f_{7+}^{(\lambda,\beta)}, \frac{1}{n}\Bigr)_p \ge C (\alpha) n^{-\alpha}
\left\| V_n^{(\alpha)} (f_{7+}^{(\lambda,\beta)})(\cdot) \right\|_p \nonumber \\
 &\ge& C (\alpha) n^{-\alpha} \left|
V_n^{(\alpha)} (f_{7+}^{(\lambda,\beta)})(0) \right|_p
 \ge C (\alpha,\beta) J_2.
 \end{eqnarray}
 By  (\ref{11}) and  $\sum\limits_{k=2n}^{\infty}  a_k    \le  4 E_n(f)_\infty$ (see
 \cite{bari-sopr}), we have
\begin{eqnarray}\label{vspom79}
\omega_\alpha\Bigl(f_{7}^{(\lambda,\beta)}, \frac{1}{n}\Bigr)_p
 \ge C (\alpha) E_{[\frac{n}{2}]}(f_{7+}^{(\lambda,\beta)})_p
 \ge C (\alpha,\beta) J_1.
 \end{eqnarray}
We have already proved that $f_7\in H_{\alpha+r}^p[\omega] \subset
W_p^{\lambda,\beta}H_\alpha[\varphi].$ Collecting inequalities (\ref{vspom78}), (\ref{vspom79}),
and (\ref{45}), we arrive at the right-hand side of (\ref{6}).
\\
{\sf Step 7(b): $p=\infty$ and $\cos \frac{\pi\alpha}{2}= 0$}. If $\cos \frac{\pi\beta}{2}\ne 0,$
then we use (\ref{vspom79}) and
\begin{eqnarray*}
\omega_\alpha\Bigl(f_{7}^{(\lambda,\beta)}, \frac{1}{n}\Bigr)_p &\ge&
 C (\alpha) n^{-\alpha}
\left\| V_n^{(\alpha)} (f_{7-}^{(\lambda,\beta)})(\cdot) \right\|_p
 \ge C (\alpha,\beta) J_2.
 \end{eqnarray*}
If $\cos \frac{\pi\beta}{2}=0,$ then $f_7=\pm f_{7+}$ and
\begin{eqnarray*}
\omega_\alpha\Bigl(f_{7}^{(\lambda,\beta)}, \frac{1}{n}\Bigr)_p
 \ge C (\alpha) E_{[\frac{n}{2}]}(f_{7+}^{(\lambda,\beta)})_p
 \ge C (\alpha,\beta) J_1.
 \end{eqnarray*}
To obtain the estimate of $J_2$, we define
\begin{equation*}
f_{10}(x) = \frac{\varepsilon_0}{2} + \left( \varepsilon_{1} - \varepsilon_{2}\right) \cos  x +
\sum\limits_{\nu=1}^\infty \left( \varepsilon_{2^{\nu}} - \varepsilon_{2^{\nu+1}} \right) \cos
2^{\nu} x.
\end{equation*}
It is clear that $E_n(f_{10})_p\le \varepsilon_{n+1}$. Then, by
 (\ref{33}), we have $f_{10}\in
H_{\alpha+r}^p[\omega] \subset
 W_p^{\lambda,\beta}H_\alpha[\varphi]$. Using
$\omega_\alpha\left(f,\frac{1}{n}\right)_p \ge C(\alpha) n^{-\alpha} \left\|  V_n^{(\alpha)} (f)
\right\|_p$
  and Lemma \ref{stlak1}, we write ($2^m\le n+1 < 2^{m+1}$)
\begin{eqnarray}\label{83}
\omega_\alpha\left(f_{10}^{(\lambda,\beta)}, \frac{1}{2^m}\right)_p &\ge& C(\alpha, \beta) 2^{ -
m\alpha}
 \sum\limits_{\nu=0}^{m}
  \lambda_{2^\nu}
2^{\nu\alpha} \left( \varepsilon_{2^{\nu}} - \varepsilon_{2^{\nu+1}} \right) \nonumber
\\
&\ge& C_1(\alpha, \beta, r) 2^{ - m\alpha}
 \sum\limits_{\nu=0}^{m}
  \lambda_{2^\nu}
2^{\nu\alpha} \varepsilon_{2^{\nu}} - C_2(\alpha, \beta, r)
  \lambda_{2^{m+1}}
\varepsilon_{2^{m+1}}.
\end{eqnarray}
Concurrently, $ \omega_\alpha\left(f_{10}^{(\lambda,\beta)}, \frac{1}{2^m}\right)_p \ge C(\alpha,
\beta) \lambda_{2^{m+1}} \varepsilon_{2^{m+1}}.$
 Then, we get
\begin{eqnarray}\label{f10}
\omega_\alpha\left(f_{10}^{(\lambda,\beta)}, \frac{1}{n}\right)_p \ge C(\alpha, \beta, r) J_2,
\end{eqnarray}
 and
by (\ref{f+-}),
\begin{eqnarray*}
C(\alpha, \beta, r) \Bigl[ J_1+J_2\Bigr]  &\le&
 \omega_\alpha\left(f_{10}^{(\lambda,\beta)},
\frac{1}{2^m}\right)_p+ \omega_\alpha\left(f_{7}^{(\lambda,\beta)}, \frac{1}{2^m}\right)_p
\\
&\asymp&
 \omega_\alpha\left(\left(f_{7}+ f_{10}\right)^{(\lambda,\beta)}, \frac{1}{n}\right)_p
= O \left[\varphi\left(\frac{1}{n}\right)\right].
\end{eqnarray*}
 The necessity in (\ref{6}) follows.
\\
{\sf Step 7(c): $p=1$ and $\cos \frac{\pi\alpha}{2}\ne 0$}.
 We use the function $f_6$. We have
$f_6\in H_{\alpha+r}^p[\omega]$ and by Lemma \ref{lemmagejt},
\begin{eqnarray*}
\omega_\alpha\Bigl(f_6^{(\lambda,\beta)}, \frac{1}{n}\Bigr)_1 &\ge& C (\alpha)
E_{n}(f_{6-}^{(\lambda,\beta)})_p \ge C (\alpha,\beta) \sum\limits_{\nu=n+1}^\infty \lambda_{\nu}
\frac{a_\nu}{\nu}
\\
&\ge&C (\alpha,\beta) \left( \sum\limits_{\nu=n+1}^\infty \lambda_{\nu}
\frac{\varepsilon_\nu}{\nu} - a_{n+1}\lambda_{n} - \sum\limits_{\nu=n+1}^\infty
\left(\lambda_{\nu}- \lambda_{\nu-1} \right) a_\nu  \right).
\end{eqnarray*}
 On the other hand,
\begin{equation*}
\omega_\alpha\Bigl(f_6^{(\lambda,\beta)}, \frac{1}{n}\Bigr)_1 \ge C (\alpha,\beta,r) \left(
a_{n+1}\lambda_{n} + \sum\limits_{\nu=n+1}^\infty \left(\lambda_{\nu}- \lambda_{\nu-1} \right)
a_\nu \right).
\end{equation*}
The last two inequalities imply $\omega_\alpha\bigl(f_6^{(\lambda,\beta)}, \frac{1}{n}\bigr)_1 \ge
C (\alpha,\beta,r) J_1$. Also,
\begin{eqnarray}\label{70}
\omega_\alpha\Bigl(f_{6}^{(\lambda,\beta)}, \frac{1}{n}\Bigr)_p &\ge&
 C (\alpha) n^{-\alpha}
\left\| V_n^{(\alpha)} (f_{6-}^{(\lambda,\beta)})(\cdot) \right\|_p
 \ge C (\alpha,\beta) J_2.
 \end{eqnarray}
\\
{\sf Step 7(d): $p=1$ and $\cos \frac{\pi\alpha}{2} =0$}.
 If $\cos \frac{\pi\beta}{2}\ne 0,$
then we use $\omega_\alpha\bigl(f_6^{(\lambda,\beta)}, \frac{1}{n}\bigr)_1 \ge C (\alpha,\beta,r)
J_1$ and
\begin{eqnarray*}
\omega_\alpha\Bigl(f_{6}^{(\lambda,\beta)}, \frac{1}{n}\Bigr)_p &\ge&
 C (\alpha) n^{-\alpha}
\left\| V_n^{(\alpha)} (f_{6+}^{(\lambda,\beta)})(\cdot) \right\|_p
 \ge C (\alpha,\beta) J_2.
 \end{eqnarray*}
If $\cos \frac{\pi\beta}{2}=0,$ we consider $f_6+f_8$. Using Lemmas \ref{my} and \ref{lemmagejt},
we have
\begin{eqnarray*}
\omega_\alpha\left(f_8^{(\lambda,\beta)}, \frac{1}{n}\right)_1 &\ge& C(\alpha, \beta) n^{-\alpha}
 \sum\limits_{\nu=1}^{n}
  \lambda_{\nu}
\nu^{\alpha-1} b_\nu
\\
&\ge& C_1(\alpha, \beta, r) n^{-\alpha}
 \sum\limits_{\nu=1}^{n}
  \lambda_{\nu}
\nu^{\alpha-1} \varepsilon_{\nu-1} - C_2(\alpha, \beta, r)
 \lambda_{n}
\varepsilon_{n}.
\end{eqnarray*}
Since $\omega_\alpha\left(f_8^{(\lambda,\beta)}, \frac{1}{n}\right)_1 \ge C(\alpha, \beta, r)
 \lambda_{n}\varepsilon_{n}$, then
$\omega_\alpha\left(f_8^{(\lambda,\beta)},\frac{1}{n}\right)_1 \ge C(\alpha, \beta,r) J_2$.
Thus
$$
C(\alpha, \beta, r) \Bigl[ J_1+J_2\Bigr]
\le
 \omega_\alpha\left(\left(f_{6}+ f_{8}\right)^{(\lambda,\beta)}, \frac{1}{n}\right)_p
=O \left[\varphi\left(\frac{1}{n}\right)\right],$$ i.e.,
 the necessity in (\ref{6}) follows.

{\sf {Step 8.} } We will prove the necessity in (\ref{6}) for the case $\sin \frac{\pi\beta}{2}=
0$. Let $H_{\alpha+r}^p[\omega] \subset W_p^{\lambda,\beta}H_\alpha[\varphi]$ and $\varepsilon$ be
a $q_{\alpha+r, \,1}(\omega)$-sequence. Since $\sin \frac{\pi\beta}{2}= 0$,  we have from
(\ref{33})
\begin{eqnarray}\label{49}
 \lambda_{n+1} \omega\left(\frac{1}{n+1}\right)
&+& n^{-\alpha} \sum\limits_{\nu=1}^n \nu^{r+\alpha} \left(\nu^{-r}\lambda_{\nu}-
(\nu+1)^{-r}\lambda_{\nu+1} \right) \omega\left(\frac{1}{\nu}\right) \nonumber
\\
&+& | \cos \frac{\beta\pi}{2} | \sum\limits_{\nu=n+2}^\infty
\left(\lambda_{\nu+1}-\lambda_{\nu}\right) \omega\left(\frac{1}{\nu}\right) \nonumber
\\
&\le& C(\alpha, \beta, r)
 \left(
\sum\limits_{\nu=n+1}^\infty
 \lambda_\nu
\left(\varepsilon_{\nu}-\varepsilon_{\nu+1}\right) + n^{-\alpha} \sum\limits_{\nu=1}^n \lambda_\nu
{\varepsilon_\nu} \nu^{\alpha-1} \right) \nonumber
\\
&=:& C(\alpha, \beta, r) \left(J_3+J_4\right).
\end{eqnarray}
\\
{\sf Step 8(a): $p=\infty$ and $\cos \frac{\pi\alpha}{2}\ne 0$}. By the Jackson inequality and
Lemma \ref{stlak1}, we have
\begin{eqnarray} \label{f102}
\omega_\alpha\left(f_{10}^{(\lambda,\beta)}, \frac{1}{2^m}\right)_p \ge C(\alpha, \beta)
\sum\limits_{\nu=m}^{\infty}  \lambda_{2^\nu} \left( \varepsilon_{2^{\nu}} -
\varepsilon_{2^{\nu+1}} \right).
\end{eqnarray}
We also note that by Lemma (\ref{stlak1}), (\ref{f10}) holds for all $\alpha>0$. This and
(\ref{f102}) give
 $\omega_\alpha\left(f_{10}^{(\lambda,\beta)}, \frac{1}{n+1}\right)_p \ge
C(\alpha, \beta) \left(J_3+J_4\right)$. Using condition (\ref{49}) and $f_{10}\in
H_{\alpha+r}^p[\omega] \subset W_p^{\lambda,\beta}H_\alpha[\varphi]$, we arrive at the relation in
the right part of (\ref{6}).
\\
{\sf Step 8(b): $p=\infty$ and $\cos \frac{\pi\alpha}{2}= 0$}. Then we consider $f_{10}$ and
$f_{11}:=\widetilde{f_{10}}$. It is clear that $f_{11}\in L_p$ and $f_{10}+f_{11}\in
H_{\alpha+r}^p[\omega]$. Nevertheless, $\omega_\alpha\left(f_{10}^{(\lambda,\beta)},
\frac{1}{n+1}\right)_p \ge C(\alpha, \beta) J_3$,
 $\omega_\alpha\left(f_{11}^{(\lambda,\beta)},
\frac{1}{n+1}\right)_p \ge C(\alpha, \beta) J_4$, and
$\omega_\alpha\left(f_{10}^{(\lambda,\beta)}, \frac{1}{n+1}\right)_p+
\omega_\alpha\left(f_{11}^{(\lambda,\beta)}, \frac{1}{n+1}\right)_p \asymp
 \omega_\alpha\left(\left(f_{10}+ f_{11}\right)^{(\lambda,\beta)}, \frac{1}{n+1}\right)_p$.
\\
{\sf Step 8(c): $p=1$ and $\cos \frac{\pi\alpha}{2}\ne 0$}. Since $f_8^{(\lambda,\beta)}(x) \sim
\pm \sum\limits_{\nu=1}^\infty
 \lambda_\nu b_\nu \sin \nu x,$ by Lemmas \ref{my} and \ref{lemmagejt}, we write (see Step 7(d))
\begin{eqnarray}\label{vspom81}
\omega_\alpha\Bigl(f_{8}^{(\lambda,\beta)}, \frac{1}{n}\Bigr)_p \ge
 C (\alpha)
 n^{-\alpha}
\left\| V_n^{(\alpha)} (f_{8-}^{(\lambda,\beta)})(\cdot) \right\|_p
 \ge C (\alpha,\beta) J_4.
 \end{eqnarray}
Also, by Lemma \ref{lemmagejt} and the Jackson inequality (\ref{11}), we  have
\begin{eqnarray*}
\omega_\alpha\left(f_8^{(\lambda,\beta)},\frac{1}{n}\right)_1 &\ge& C(\alpha, \beta)
 \sum\limits_{\nu=n+1}^{\infty}
  \lambda_{\nu} \frac{b_\nu}{\nu}
\nonumber
\\
&\ge& C(\alpha, \beta)
 \sum\limits_{\nu=n+1}^{\infty}
  \lambda_{\nu}
\sum\limits_{j=2\nu-1}^\infty \frac{\varepsilon_j - \varepsilon_{j+1}}{j+1} \nonumber
\\
&\ge& C(\alpha, \beta, r) \sum\limits_{j=4n-1}^\infty \left(\varepsilon_j -
\varepsilon_{j+1}\right)
  \lambda_{\nu} .
\end{eqnarray*}
Using the properties of modulus of smoothness, we get (\ref{6}).
\\
{\sf Step 8(d): $p=1$ and $\cos \frac{\pi\alpha}{2}= 0$}. We use that  $f_{12}:=\widetilde{f_8}
\in L_1$ and $E_n(f_{12})_p\le C \varepsilon_{n+1}$ (see \cite{gejt}). Then $f_8+f_{12}\in
H_{\alpha+r}^p[\omega]$ and $\omega_\alpha\left(f_8^{(\lambda,\beta)},\frac{1}{n}\right)_1 \ge
C(\alpha, \beta) J_3$, $\omega_\alpha\left(f_{12}^{(\lambda,\beta)},\frac{1}{n}\right)_1 \ge
C(\alpha, \beta) J_4$. This completes the proof of Theorem 1.






\section{Corollaries. Estimates of transformed Fourier series.}
Theorem 1 actually provides estimates of the norms and moduli of smoothness of the transformed
Fourier series, i.e., the estimates of $\|\varphi\|_p$ and $\omega_\alpha(\varphi,\delta)_p$,
where $\varphi \sim \sigma(f, \lambda)$, by  $\omega_\gamma(f,\delta)_p$. Analyzing Theorem 1, one
can see that the  following two conditions play a crucial role. The first
 is the behavior of the transforming sequence $\{\lambda_n\}$ and the second is the alternation between $L_p,$
$1<p<\infty$ and $L_p,$ $p=1, \infty$.

We will investigate in detail some important examples
 for  $L_p,$ $1<p<\infty$  and  for $L_p,$ $p=1, \infty$, separately.
\\
\begin{center}
{\bf 1. The case  ${\bf 1<p<\infty}$. }
\end{center}
\begin{thr} \label{thr1}
 Let\, $1<p<\infty$, $\theta=\min (2,p)$
$\tau=\max (2,p)$, $\alpha\in  \mathbf{R}_+,$ and $\lambda=\left\{\lambda_n\right\}$ be a
non-decreasing sequence of positive numbers. Let  $\rho$  be a non-negative number such that the
sequence $\left\{n^{-\rho}\lambda_n\right\}$ is non-increasing.
\\
{\bf I. }\,\, If for $f\in L_p^0$ the series
$$
\sum\limits_{n=1}^\infty  \left(\lambda_{n+1}^\theta-\lambda_{n}^\theta\right) \omega^\theta
_{\alpha+\rho}\left(f, \frac{1}{n}\right)_p\,$$ converges, then there exists a function
$\varphi\in L_p^0$ with Fourier series $ \sigma(f, \lambda)$, and
\begin{eqnarray}
\label{t1} \|\varphi\|_p &\le& C(p,\lambda, \alpha, \rho) \left\{
\lambda_1^\theta \|f\|_p^\theta + \sum\limits_{n=1}^\infty
\left(\lambda_{n+1}^\theta-\lambda_{n}^\theta\right) \omega^\theta
_{\alpha+\rho}\left(f, \frac{1}{n}\right)_p
\right\}^\frac{1}{\theta},
\\
\omega _{\alpha}\left(\varphi, \frac{1}{n+1}\right)_p &\le&
C(p,\lambda, \alpha, \rho) \left\{ n^{-\alpha\theta}
\sum\limits_{\nu=1}^n \nu^{(\rho+\alpha)\theta}
\left(\nu^{-\rho\theta}\lambda_{\nu}^\theta-
(\nu+1)^{-\rho\theta}\lambda_{\nu+1}^\theta \right) \omega^\theta
_{\alpha+\rho}\left(f, \frac{1}{\nu}\right)_p \right. \nonumber
\\
&+&
\sum\limits_{\nu=n+2}^\infty
\left(\lambda_{n+1}^\theta-\lambda_{n}^\theta\right)
\omega^\theta _{\alpha+\rho}\left(f, \frac{1}{\nu}\right)_p
+
\left.
\lambda_{n+1}^\theta
\omega^\theta _{\alpha+\rho}\left(f, \frac{1}{n}\right)_p
\right\}^\frac{1}{\theta}.\label{t2}
\end{eqnarray}
{\bf II. } \,\,If for $f\in L_p^0$ there exists a function $\varphi\in L_p$ with Fourier series
$\sigma(f, \lambda)$, then
\begin{eqnarray}
\label{r5} \left\{ \lambda_1^\tau \|f\|_p^\tau +
\sum\limits_{n=1}^\infty \left(\lambda_{n+1}^\tau-\lambda_{n}^\tau
\right) \omega^\tau _{\alpha+\rho}\left(f, \frac{1}{n}\right)_p
\right\}^\frac{1}{\tau} &\le& C(p,\lambda, \alpha, \rho)
\|\varphi\|_p,
\\
\left\{
n^{-\alpha\tau}
\sum\limits_{\nu=1}^n
\nu^{(\rho+\alpha)\tau}
\left(\nu^{-\rho\tau}\lambda_{\nu}^\tau-
(\nu+1)^{-\rho\tau}\lambda_{\nu+1}^\tau \right)
\omega^\tau _{\alpha+\rho}\left(f, \frac{1}{\nu}\right)_p
\right.
&+&\label{r6}
\\
\sum\limits_{\nu=n+2}^\infty
\left(\lambda_{n+1}^\tau-\lambda_{n}^\tau\right) \omega^\tau
_{\alpha+\rho}\left(f, \frac{1}{\nu}\right)_p +\left.
\lambda_{n+1}^\tau \omega^\tau_{\alpha+\rho}\left(f,
\frac{1}{n}\right)_p \right\}^\frac{1}{\tau} &\le& C(p,\lambda,
\alpha, \rho) \omega_{\alpha}\left(\varphi,
\frac{1}{n+1}\right)_p, \nonumber
\end{eqnarray}
\begin{eqnarray}
\label{tt3} \omega_{\alpha+\rho}\left(f, \frac{1}{n}\right)_p
&\le& C(p,\lambda, \alpha, \rho)
\frac{\|\varphi\|_p}{\lambda_{n}},
\\
\label{tt4} \omega_{\alpha+\rho}\left(f, \frac{1}{n}\right)_p
&\le& C(p,\lambda, \alpha, \rho)
\frac{\omega_{\alpha}\left(\varphi, \frac{1}{n}\right)_p}
{\lambda_{n}}.
\end{eqnarray}
\end{thr}
Inequalities (\ref{t1})-(\ref{t2}) and (\ref{tt3})-(\ref{tt4}) were proved in Theorem 1 (see the
sufficiency in part {\bf I}). To prove (\ref{r5})-(\ref{r6}), we use similar reasoning, applied to
the theorems by Littlewood-Paley,  Marcinkiewicz, and the Minkowski's inequality.

As an important corollary of Theorem \ref{thr1},

\begin{con} \label{con1}
 Let\, $1<p<\infty$, $\theta=\min (2,p)$
$\tau=\max (2,p)$. Then, for any $k, r>0$,
\begin{multline}
\label{b3} C_1 \left\{ \sum\limits_{\nu=n+1}^\infty \nu^{r\tau-1}
\omega_{k+r}^\tau \left(f,\frac{1}{\nu}\right)_p
\right\}^\frac{1}{\tau} \le \omega_{k}
\left(f^{(r)},\frac{1}{n}\right)_p \le  C_2\left\{
\sum\limits_{\nu=n+1}^\infty \nu^{r\theta-1} \omega_{k+r}^\theta
\left(f,\frac{1}{\nu}\right)_p \right\}^\frac{1}{\theta},
\end{multline}
where $C_1=C_1(p,k,r), C_2=C_2(p,k,r), n \in \mathbf{N}.$
\end{con}
The last two inequalities are an improvement compared to (\ref{b1}) and (\ref{b2}). Indeed, by
 properties of the modulus of smoothness and by the Jensen inequality,
\begin{eqnarray*}
& &\qquad n^r \omega_{k+r} \left(f,\frac{1}{n}\right)_p \le C(k,r)
 \left\{\sum\limits_{\nu=n+1}^\infty
\nu^{r\tau-1} \omega_{k+r}^\tau \left(f,\frac{1}{\nu}\right)_p \right\}^\frac{1}{\tau} ;
\\
& & \left\{ \sum\limits_{\nu=n+1}^\infty \nu^{r\theta-1} \omega_{k+r}^\theta
\left(f,\frac{1}{\nu}\right)_p \right\}^\frac{1}{\theta}
 \le C(k,r) \sum\limits_{\nu=n+1}^\infty
\nu^{r-1} \omega_{k+r} \left(f,\frac{1}{\nu}\right)_p.
\end{eqnarray*}
{\bf Example. } Let $\psi(t)=t^r  \ln^{-A} (1/t)$ and $2\le p <\infty, \frac{1}{2}<A<1.$ For
$\omega_{k+r} \left(f,t\right)_p \asymp \psi(t)$,
(\ref{b1}) and (\ref{b2}) give only $C \ln^{-A}
(1/t) \le \omega_{k} \left(f^{(r)}, t \right)_p$, and  (\ref{b3}) gives $C_1 \ln^{-A+1/p} (1/t)
\le   \omega_{k} \left(f^{(r)}, t \right)_p \le C_2 \ln^{-A+1/2} (1/t) $, which is superior.

Proof of Corollary \ref{con1} is obvious from (\ref{t2}) and (\ref{r6}) with $r=\rho$, because if
$f\in L_p,$ $1<p<\infty$, then $f^{(r)}\sim \sigma(f, \lambda)$ for $\{\lambda_n=n^r\}$.

From  (\ref{b1}), (\ref{b2}) and  (\ref{b3}), one can see that it is natural to estimate
$\omega_{\alpha} \left(f^{(\gamma)},\delta\right)_p$ by $\omega_{\alpha+r} \left(f,t\right)_p$.
Further analysis allows the distinction between three different types of such estimates. It will
be convenient for us to write inequalities in integral form:


1. $\gamma=r$ (see  Corollary \ref{con1})\footnote{ Here and
further $\tau=\max(2,p), \theta=\min(2,p)$. We will write $A_1\ll
A_2$, if $A_1\le C A_2$, $C\ge 1$. Also, if $A_1\ll A_2$ and
$A_2\ll A_1$, then $A_1\asymp A_2$.}
\begin{equation} \label{weyl1}
\left\{ \int\limits_{0}^{\delta} t^{-r{\tau}-1}
\omega^{\tau}_{r+\alpha}(f,t)_p
 dt \right\}^{\frac{1}{\tau}} \ll \omega_\alpha(f^{(r)},\delta)_p \ll
\left\{ \int\limits_{0}^{\delta} t^{-r{\theta}-1} \omega^{\theta}_{r+\alpha}(f,t)_p
 dt \right\}^{\frac{1}{\theta}};
\end{equation}


2. $\gamma=r-\varepsilon,$\, $0<\varepsilon<r$
(see Theorem \ref{thr1} for $\rho=r$ and $\lambda_n=n^{r-\varepsilon}$):
\begin{equation} \label{weyl4}
 \left\{ \int\limits_{0}^{\delta} t^{-(r-\varepsilon){\tau}-1}
 \omega^{\tau}_{r+\alpha}(f,t)_p
 dt + \delta^{\alpha \tau}
\int\limits_{\delta}^{1}  t^{-(r-\varepsilon+\alpha){\tau}-1}
 \omega^{\tau}_{r+\alpha}(f,t)_p dt
 \right\}^{\frac{1}{\tau}}  \ll \omega_\alpha(f^{(r-\varepsilon)},\delta)_p,
\end{equation}
\begin{equation}\label{weyl5}
 \omega_\alpha(f^{(r-\varepsilon)},\delta)_p \ll
\left\{ \int\limits_{0}^{\delta} t^{-(r-\varepsilon){\theta}-1}
 \omega^{\theta}_{r+\alpha}(f,t)_p
 dt + \delta^{\alpha \theta}
\int\limits_{\delta}^{1}  t^{-(r-\varepsilon+\alpha){\theta}-1}
 \omega^{\theta}_{r+\alpha}(f,t)_p dt
 \right\}^{\frac{1}{\theta}};
\end{equation}


3. $\gamma=r+\varepsilon,$\, $0<\varepsilon<\alpha,$
(see \cite{2003}):
\begin{equation} \label{weyl2}
\left\{ \int\limits_{0}^{\delta} t^{-(r+\varepsilon){\tau}-1}
 \omega^{\tau}_{r+\alpha}(f,t)_p  \,dt\right\}^{\frac{1}{\tau}} \ll
 \delta^{\alpha-\varepsilon}\left\{
\int\limits_{\delta}^{1}  t^{-(\alpha-\varepsilon){\theta}-1}
\omega^{\theta}_\alpha(f^{(r+\varepsilon)},t)_p\, dt
\right\}^{\frac{1}{\theta}},
\end{equation}
\begin{equation} \label{weyl3}
 \delta^{\alpha-\varepsilon}\left\{
\int\limits_{\delta}^{1}  t^{-(\alpha-\varepsilon){\tau}-1}
\omega^{\tau}_\alpha(f^{(r+\varepsilon)},t)_p dt \right\}^{\frac{1}{\tau}}
\ll
\left\{ \int\limits_{0}^{\delta} t^{-(r+\varepsilon){\theta}-1}
 \omega^{\theta}_{r+\alpha}(f,t)_p  dt\right\}^{\frac{1}{\theta}}.
\end{equation}
One can consider a more general sequence than $\{\lambda_n=n^\gamma\}$, $\gamma>0$. Let us define
$\{\Lambda_n(s):= \Lambda(s,\frac{1}{n})\}$, where
\begin{equation} \label{posl}
\Lambda(s,t)=\Lambda(s,r,t)=\left( \int\limits^1_t  \xi(u) du +
t^{-rs}\int\limits^t_0 u^{rs} \xi(u) du  \right)^\frac{1}{s},
\end{equation}
and non-negative function $\xi(u)$ on  $[0,1]$ is such that
$u^{rs} \xi(u)$ is summable.


The sequence $\{\Lambda_n=\Lambda_n(s)\}$ is often considered a  transforming  sequence. \\
{\bf Example. } 1. $\{\Lambda_n=n^\gamma\}$, $0<\gamma<r$; 2.
$\{\Lambda_n=n^{r} \ln^{-A} n\}$, $A>0$; 3. $\{\Lambda_n= \ln^{A}
n\}$, $A>0$.

Properties of  $\Lambda(s,t)$: $\Lambda(s,t)$ is non-increasing on $t$,
 $\Lambda(s,r,t) t^{r}$ is non-decreasing on $t$.
By these properties and  Theorem \ref{thr1}, one can obtain the following estimates, which are
more general than (\ref{weyl4})--(\ref{weyl5})
 (see also \cite{ps1}, \cite{1997st}).

\begin{thr} \label{thr2}
 Let  $1<p<\infty$, $\theta=\min (2,p)$
$\tau=\max (2,p)$, $\alpha, r>0$, and
 $\Lambda=\{\Lambda_n(s): = \Lambda(s,\frac{1}{n})\}$, where
$\Lambda(s,t)$ and $\xi(t)$ were denoted above.
\\
{\bf I.} \,\,
If for $f\in L_p$ the integral
$$I_1:=  \left\{
\int\limits^1_0  \xi(t) \,\, \omega^s_{\alpha+r}\left(f, t\right)_p   \,dt \right\}^\frac{1}{s} \,
$$ is finite for $s= \theta$, then there exists a function $\varphi\in L_p$
with Fourier series
$\sigma(f, \Lambda)$,
 $\|\varphi\|_p \le C(p, \xi, \alpha, r, s) I_1$,
and
\begin{eqnarray}
\label{tt1}
\omega_{\alpha}\left(\varphi, \delta \right)_p
&\le& C(p, \xi, \alpha, r, s)
\left\{
 \int\limits^\delta_0  t^{-r s-1} \,
\int\limits^t_0 u^{r s} \xi(u)\, du \,
 \omega_{r+\alpha}^s(f,t)\, dt
\right.
\nonumber
\\
&+&
\left.
\delta^{\alpha s}
 \int\limits_\delta^1  t^{\alpha s-1} \,
\int\limits_t^1  \xi(u)\, du \,
 \omega_{r+\alpha}^s(f,t)_p\, dt \right\}^\frac{1}{s} =:C \,I_2.
\end{eqnarray}
{\bf II. } \,\,If for $f\in L_p$ there exists a function $\varphi\in L_p$
with Fourier series   $\sigma(f, \Lambda)$, then for $s=\tau$
$\|\varphi\|_p \ge C(p, \xi, \alpha, r, s) I_1$, and
$\omega_{\alpha}\left(\varphi, \delta \right)_p
\ge C(p, \xi, \alpha, r, s) I_2.$
\end{thr}
In a similar way we can generalize inequalities (\ref{weyl2})--(\ref{weyl3}).
\begin{thr} \label{thr3}
Let   $1<p<\infty$, $\theta=\min (2,p)$
$\tau=\max (2,p)$, $\alpha, r>0$,  and
 $\Lambda=\{\Lambda_n(s): = \Lambda(s,\frac{1}{n})\}$, where
$\Lambda(s,t)$ and $\xi(t)$ were denoted above.
\\
{\bf I. } \,\,
If for $f\in L_p$ the integral
$$
I_3:= \left\{ \int\limits^1_0  \xi(t) \,\, \omega^s_{\alpha+r}\left(f, t\right)_p   \,dt
\right\}^\frac{1}{s} \, $$ is finite for $s= \theta$, then there exists a function $\varphi\in
L_p$ with Fourier series $\sigma(f, \Lambda)$,
 $\|\varphi\|_p \le C(p, \xi, \alpha, r, s) I_3$,
and for any $\alpha_1>\alpha$
\begin{eqnarray}
\label{ttt1}
I_4(\tau, s)=: \left\{ \delta^{(r+\alpha)\tau} \Lambda^\tau(s,r,\delta)
 \int\limits_\delta^1 \frac{ t^{-(r+\alpha) \tau-1}}
{\Lambda^\tau(s,r,t)}
 \omega_{r+\alpha_1}^\tau(\varphi,t)_p\, dt \right\}^\frac{1}{\tau}
 \le
C(p,\xi, \alpha, \alpha_1, r, s) \nonumber
\\
\times
 \left[
 \Lambda(s,r,\delta) \omega_{r+\alpha}(f,\delta)_p
+
 \left\{
 \int\limits^\delta_0  t^{-r \theta-1} \,
\int\limits^t_0 u^{r \tau} \xi(u)\, du \,
 \omega_{r+\alpha}^\theta(f,t)_p\, dt \right\}^\frac{1}{\theta}
 \right]=:
 C I_5(\theta)
\end{eqnarray}
{\bf II. } \,\,If for $f\in L_p$ there exists a function $\varphi\in L_p$ with Fourier series
$\sigma(f, \Lambda)$, then for $s=\tau$ and for any  $\alpha_1>\alpha$ $\|\varphi\|_p \ge
C(p,\lambda, \alpha, r, s) I_3$ and $I_4(s, \theta)\ge C(p,\xi, \alpha, \alpha_1, r, s)  I_5(s)$.
\end{thr}
\begin{center}
{\bf 2. The case  ${\bf p=1, \infty}$. }
\end{center}
 Estimates $\omega_{\alpha}(\varphi,t)_p$ by $\omega_{r+\alpha}(f,t)_p$
for this case follow from Theorem 1  (see part {\bf II}). We will
write only the commonly used estimates  of \,$\omega_{\alpha}
(f^{(r)},t)_p$ and $\omega_{\alpha}(\tilde{f}^{(r)},t)_p$ by
$\omega_{r+\alpha}(f,t)_p$.

\begin{con} \label{con2}
If  $p=1,\infty$, then  (\ref{b1}), (\ref{b2}) are true for any
 $k, r>0.$
\end{con}
As a corollary of Theorem 1 for
$\left\{\lambda_n=n^\rho\right\}$, $\rho\ge 0$ and $\beta=\rho+1$,
\begin{con} \label{con25}
Let $p=1,\infty$. One has
$$
H_{\alpha+\rho}^p[\omega] \subset
\widetilde{W}^\rho_p
\Longleftrightarrow
\sum\limits_{n=1}^\infty
n^{\rho-1} \omega\left(\frac{1}{n}\right)\, < \,\infty.
$$
\end{con}
For $p=1, \rho=0$ and $\alpha=1$ Corollary \ref{con25} gives the answer for the question by F.
Moricz \cite{mo}. Also, we mention the papers \cite{bern}, \cite{hass}, \cite{stechkin}, where the
embedding results were obtained in the necessity part.



\begin{con} \label{con3}
Let $p=1,\infty$, and  $r, \alpha, \varepsilon>0$.
\\
{\bf I. }\,\, If for $f\in L_p$ the series $\sum\limits_{\nu=1}^\infty \nu^{r-1}
\omega_{r+\alpha+\varepsilon} \left(f,\frac{1}{\nu}\right)_p$ converges, then there exists
$\tilde{f}^{(r)} \in L_p$, and
\begin{equation*}
\omega_{\alpha} \Bigl(\widetilde{f}^{(r)},\frac{1}{n}\Bigr)_p \le  C(r,\alpha, \varepsilon)
\left(
n^{-\alpha}
\sum\limits_{\nu=1}^n \nu^{r+\alpha-1}
\omega_{r+\alpha+\varepsilon}\Bigl(f,\frac{1}{\nu}\Bigr)_p
+ \sum\limits_{\nu=n+1}^\infty \nu^{r-1}
\omega_{r+\alpha+\varepsilon} \Bigl(f,\frac{1}{\nu}\Bigr)_p
\right), \,\, n \in \mathbf{N}.
\end{equation*}
{\bf II. } \,\,If for $f\in L_p$ there exists $\tilde{f}^{(r)}\in L_p$,
then
\begin{equation} \label{b5}
\omega_{r+\alpha+\varepsilon} \left(f,\frac{1}{n}\right)_p \le
\frac{C(r, \alpha, \varepsilon)}{n^r}
\omega_{\alpha} \left(\tilde{f}^{(r)},\frac{1}{n}\right)_p,
 \qquad n \in \mathbf{N}.
\end{equation}
\end{con}
Using direct and inverse approximation theorems, we can
rewrite inequality in {\bf I } in the following equivalent form
(compare to \cite{ba-st}):
\begin{con} \label{con35}
Let $p=1,\infty$, and $r, \alpha>0$.
If for $f\in L_p$ the series
$\sum\limits_{\nu=1}^\infty \nu^{r-1}  E_\nu\left(f\right)_p$
converges, then there exists
$\tilde{f}^{(r)} \in L_p$, and
\begin{equation*}
\omega_{\alpha} \Bigl(\widetilde{f}^{(r)},\frac{1}{n}\Bigr)_p \le  C(r,\alpha, \varepsilon)
\left(
n^{-\alpha}
\sum\limits_{\nu=1}^n \nu^{r+\alpha-1}  E_\nu\left(f\right)_p
+ \sum\limits_{\nu=n+1}^\infty \nu^{r-1}
E_\nu\left(f\right)_p \right), \qquad n \in \mathbf{N}.
\end{equation*}
\end{con}




\section{Final remarks}


1. The Weyl class  $W_p^{\lambda, \beta}$ coincides with the class of functions from $L(0, 2\pi)$
such that their Fourier series can be presented in the following form
$$
\frac{a_0 (f)}{2}+
\sum\limits_{\nu=1}^\infty
 \frac{1}{\pi\lambda_\nu}
\int\limits_0^{2\pi} \psi(x-t) \cos \Bigl(\nu t-\frac{\pi\beta}{2}\Bigr)\,dt,\quad
\psi(x)\in L^0.
$$
Further, consider the case when
$$\sum\limits_{\nu=1}^\infty   \frac{1}{\lambda_\nu}
\cos \Bigl(\nu t-\frac{\pi\beta}{2}\Bigr)$$ is the Fourier series of a summable function
$D_{\lambda,\beta}(t)$. For example, it is so if  $\{\lambda_\nu \uparrow \infty\}$ ($n\uparrow$),
 and $\sum\limits_{\nu=1}^\infty  \frac{1}{\nu\lambda_\nu}<\infty$.
Then the elements of $W_p^{\lambda, \beta}$ can differ only by the
mean value from functions $f$, which have the following
representation by convolution,
$$f(x)=  \frac{1}{\pi}
\int\limits_0^{2\pi} \psi(x-t)D_{\lambda,\beta}(t) \,dt,\quad \psi(x)\in L^0.
$$
Here,  $\psi$ coincides almost everywhere with
$f^{(\lambda,\beta)}.$ See, for example, \cite[Ch. 11]{butzbook}.
A representation by convolution was first considered in \cite{sz}.
\\
2. The generalization of ${W}^r_p E[\xi]$ is the class
\begin{equation*}
{W}^{\lambda, \beta}_pE[\xi]
=
\Bigl\{f\in W_p^{\lambda, \beta} :\,\,\,
E_n \left( f^{(\lambda,\beta)} \right)_p=
O\left[\xi(1/n)\right] \Bigr\}.
\end{equation*}
We do not consider in this paper the embedding results between ${W}^{\lambda, \beta}_p E[\xi]$,
${W}^{\lambda, \beta}_pH_\alpha[\varphi]$ and $E[\varphi]\equiv {W}^{\{1\}, \,0}_p E[\varphi]$. We
only  notice that some results  of  such types follow from direct and inverse theorems
(\ref{a1})-(\ref{a4}). Also, we mention the papers \cite{h-l} and \cite{og}.
\\
3. The results of sections 2 and 5 can be generalized to a multidimensional case. We only write
here the following estimates for the mixed modulus  of smoothness  $\omega_{\alpha_1,\alpha_2}
(f,\delta_1,\delta_2)_p$
 of a function~$f$ (in the $L_p$ metric) of orders $\alpha_1$ and $\alpha_2$ ($\alpha_1,\alpha_2>0$)
with respect to the variables $x_1$ and~$x_2$, respectively (see, for example, \cite{2003}).
\begin{thr} \label{t5}
Let  $f(x_1,x_2)\in L_p^0$, $1<p< \infty$, $\theta= \min(2,p),  \tau=\max(2,p)$, and let
  $\alpha_1, \alpha_2$, $r_1, r_2>0$,
\\
{\bf I.} \,\,If
\begin{equation*} \label{th01}
J_1(\theta):= \left( \int\limits^{1}_{0}\int\limits^{1}_{0}
t_1^{-r_1\theta-1} t_2^{-r_2\theta-1} \omega^\theta_{r_1+\alpha_1,
r_2+\alpha_2}(f,t_1, t_2)_p \,dt_1\, dt_2 \right)^\frac{1}{\theta}
\,<\infty,
\end{equation*}
then $f$ has a mixed derivative in the sense of Weyl $f^{(r_1,r_2)}\in L_p^0$,
$$\|f^{(r_1,r_2)}\|_p\le  C(p,r_1,r_2)J_1(\theta),$$ and
\begin{multline*} \label{th01}
\omega_{\alpha_1, \alpha_2} (f^{(r_1,r_2)},\delta_1, \delta_2)_p
\le C(p,\alpha_1, \alpha_2,r_1,r_2) \left(
\int\limits_{0}^{\delta_1}\int\limits_{0}^{\delta_2}
t_1^{-r_1\theta-1} t_2^{-r_2\theta-1} \omega^\theta_{r_1+\alpha_1,
r_2+\alpha_2}(f,t_1, t_2)_p \,dt_1\,  dt_2
\right)^\frac{1}{\theta}
\\=:C\, J_2(\theta).
\end{multline*}
{\bf II.}  \,\,If $f$ has a mixed derivative
in the sense of Weyl $f^{(r_1,r_2)}\in L_p^0$, then
 $$J_1(\tau)\le  C(p,r_1,r_2)  \|f^{(r_1,r_2)}\|_p,$$ and
$$ J_2(\tau)
 \le   C(p,\alpha_1, \alpha_2,r_1,r_2)
 \omega_{\alpha_1, \alpha_2}(f^{(r_1,r_2)},\delta_1, \delta_2)_p.
$$
\end{thr}

\begin{thr} \label{t4}
Let $f(x_1,x_2)\in L_p^0$, $p=1, \infty$, and let $\alpha_1, \alpha_2, r_1, r_2>0$.
\\
{\bf I.}  \,\,
If
\begin{equation*} \label{th01}
J_3:= \int\limits^{1}_{0}\int\limits^{1}_{0}
t_1^{-r_1-1} t_2^{-r_2-1}
\omega_{r_1+\alpha_1, r_2+\alpha_2}(f,t_1, t_2)_p \,dt_1\,  dt_2  \,<\infty,
\end{equation*}
then $f$ has a mixed derivative in the sense of Weyl $f^{(r_1,r_2)}\in L_p^0$,
$$\|f^{(r_1,r_2)}\|_p \le C(r_1,r_2) J_3,$$
 and
\begin{equation*} \label{th01}
\omega_{\alpha_1, \alpha_2} (f^{(r_1,r_2)},\delta_1, \delta_2)_p
\le C(\alpha_1, \alpha_2,r_1,r_2)
\int\limits_{0}^{\delta_1}\int\limits_{0}^{\delta_2}
t_1^{-r_1-1} t_2^{-r_2-1}
 \omega_{r_1+\alpha_1, r_2+\alpha_2}(f,t_1, t_2)_p \,dt_1\,  dt_2.
\end{equation*}
{\bf II.} \,\,If $f$ has a mixed derivative
in the sense of Weyl $f^{(r_1,r_2)}\in L_p^0$, then
$$
\omega_{r_1+\alpha_1, r_2+\alpha_2}(f,\delta_1, \delta_2)_p
\le C(\alpha_1, \alpha_2,r_1,r_2)
\delta_1^{r_1} \delta_2^{r_2} \|f^{(r_1,r_2)}\|_p, $$ and
\begin{equation*} \label{th02}
\omega_{r_1+\alpha_1, r_2+\alpha_2}(f,\delta_1, \delta_2)_p
\le
  C(\alpha_1, \alpha_2,r_1,r_2)
\delta_1^{r_1} \delta_2^{r_2}
 \omega_{\alpha_1, \alpha_2} (f^{(r_1,r_2)},\delta_1, \delta_2)_p.
\end{equation*}
\end{thr}
See \cite{2003} for the  estimates of transformed series in a multidimensional case.
\\
4. In view of inequalities (\ref{weyl1}) and (\ref{weyl4}) -
(\ref{weyl3}), the problem of finding the estimates of
$\omega_{\alpha}\left(\varphi, t\right)_p$ by
$\omega_{\alpha+r}\left(f, t\right)_p$ arises such as in the case
$\varphi\sim \sigma(f, \lambda)$ with $\lambda_n=n^r \ln^A n$. If
$A<0$ (that is the analogue of the case
$\lambda_n=n^{r-\varepsilon}$), then estimates
$\omega_{\alpha}\left(\varphi, t\right)_p$ follow from Theorem
\ref{thr2}. For example, if $p=2$ and $\varphi\sim \sigma(f, n^r
\ln^A n), A<0$, then
\begin{equation} \label{hhj}
\omega^2_\beta(\varphi,\delta)_2
\asymp
 \int\limits_{0}^{\delta} \frac{t^{-2r-1}}{\ln^{2|A|}\left(\frac{2}{t}\right)}
 \omega^2_{r+\beta}(f,t)_2\, dt + \delta^{2\beta}
\int\limits_{\delta}^{1} \frac{t^{-2(r+\beta)-1}}
{\ln^{1+2|A|}\left(\frac{2}{t}\right)}
 \omega^2_{r+\beta}(f,t)_2 \,dt.
\end{equation}
 Note that the only
difference between (\ref{hhj}) and (\ref{weyl4})-(\ref{weyl5}) is related
to the replacement of $n^{-\varepsilon}$ by  $\ln^A n$.

The case $A>0$ (that is the analogue of the case
$\lambda_n=n^{r+\varepsilon}$) is interesting.
For $p=2$ and $\varphi\sim \sigma(f, n^r \ln^A n), A>0$ we have
\begin{equation}\label{hj}
\delta^{2\beta} \ln^{2A}\left(\frac{2}{\delta}\right)
\int\limits_{\delta}^{1} \frac{t^{-2\beta-1}}{
\ln^{1+2A}\left(\frac{2}{t}\right)}
\omega^2_\beta(\varphi,t)_2 \, dt
+\omega^2_\beta(\varphi,\delta)_2
\asymp
 \int\limits_{0}^{\delta} t^{-2r-1}
\ln^{2A}\left(\frac{2}{t}\right) \omega^2_{r+\beta}(f,t)_2
\, dt.
\end{equation}
Comparing this with  the estimates (\ref{weyl2})-(\ref{weyl3}) one can remark that the new item
$\omega^2_\beta(\varphi,\delta)_2$ has appeared in (\ref{hj}).
Thus, this case has essential
distinctions. See for detail \cite{2004}, \cite{tikh}.
\\
5. Defining the class  $W_p^{\lambda, \beta} H_\alpha[\varphi]$ we assumed that
 $\varphi\in \Phi_\alpha$. This restriction is natural for  a majorant
of the modulus of smoothness of order $\alpha$ (see \cite{tikh-dr}).
\\
6. In Theorem 1 (item II) we used the inequality $\sum\limits_{\nu=n+1}^\infty
\frac{1}{\nu\lambda_\nu} \le \frac{C}{\lambda_n}$. We remark that it is equivalent to the
following condition: there exists  $\varepsilon>0$ such that the sequence
 $\left\{n^{-\varepsilon}\lambda_n\right\}$ is almost increasing, i.e.,
$n^{-\varepsilon}\lambda_n\le C m^{-\varepsilon}\lambda_m$, $C\ge
1$, $n\le m.$ This and other equivalence results can be found in
\cite{ba-st} and \cite{ti}.



\section{Acknowledgements}
The first author acknowledges financial support from the Russian Fund of Basic Research  (project
03-01-00080). The second author acknowledges financial support from the European Commission
(Contract MIF1-CT-2004-509465). The paper were partially written while the authors were staying at
the Centre de Recerca Matematica (CRM) in Barcelona, Spain.


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\vspace{6mm}
\begin{raggedright}
BORIS SIMONOV
\\
VOLGOGRAD STATE TECHNICAL UNIVERSITY
\\ VOLGOGRAD, 400131
\\ RUSSIA
\\
e-mail: {\tt htf@vstu.ru}

\vspace{7mm}
 SERGEY TIKHONOV
\\
 CENTRE DE RECERCA MATEMATICA (CRM)
\\
BELLATERRA (BARCELONA) E-08193,
 SPAIN
\\
e-mail: {\tt stikhonov@crm.es}\\
\end{raggedright}




\end{document}

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