Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Weighted moduli of smoothness of $k$-monotone functions and applications   Let $\omega_\varphi^k(f,\delta)_{w,L_q}$ be the Ditzian-Totik modulus with
weight $w$, $M^k$ be the cone of $k$-monotone functions on $(-1,1)$, i.e.,
those functions whose $k$th divided differences are nonnegative for all
selections of $k+1$ distinct points in $(-1,1)$, and denote $E (X, \Pi_n)_{w,q}
:= \sup_{f\in X} \inf_{P\in\Pi_n}\|w(f-P)\|_{L_q}$, where $\Pi_n$ is the set of
algebraic polynomials of degree at most $n$. Additionally, let
$w_{\alpha,\beta}(x) := (1+x)^\alpha (1-x)^\beta$ be the classical Jacobi
weight, and denote by $S_p^{\alpha,\beta}$ the class of all functions such that
$\| w_{\alpha,\beta}f\|_{L_p}=1$.
  In this paper, we determine the exact behavior (in terms of $\delta$) of
$\sup_{f\in S_p^{\alpha,\beta}\cap M^k}
\omega_\varphi^k(f,\delta)_{w_{\alpha,\beta},L_q}$ for $1\leq p, q\leq \infty$
(the interesting case being $q<p$ as expected) and $\alpha,\beta >-1/p$ (if
$p<\infty$) or $\alpha,\beta\geq 0$ (if $p=\infty$). It is interesting to note
that, in one case, the behavior is different for $\alpha=\beta=0$ and for
$(\alpha,\beta)\neq (0,0)$. Several applications are given. For example, we
determine the exact (in some sense) behavior of $E (M^k\cap S_p^{\alpha,\beta},
\Pi_n)_{w_{\alpha,\beta},L_q}$ for $\alpha,\beta \geq 0$.

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The categories of this paper are: hep-ex
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The categories of this paper are: eess
37
The categories of this paper are: supr-con
11
The categories of this paper are: hep-lat
4
The categories of this paper are: cond-mat
36
The categories of this paper are: cmp-lg
34
The categories of this paper are: chem-ph
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The categories of this paper are: nucl-th
10
The categories of this paper are: q-bio
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The categories of this paper are: adap-org
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The categories of this paper are: nlin
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The categories of this paper are: econ
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The categories of this paper are: stat
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The categories of this paper are: funct-an
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The categories of this paper are: cs
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The categories of this paper are: comp-gas