Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Hilbert points in Hardy spaces   A Hilbert point in $H^p(\mathbb{T}^d)$, for $d\geq1$ and $1\leq p \leq
\infty$, is a nontrivial function $\varphi$ in $H^p(\mathbb{T}^d)$ such that
$\| \varphi \|_{H^p(\mathbb{T}^d)} \leq \|\varphi + f\|_{H^p(\mathbb{T}^d)}$
whenever $f$ is in $H^p(\mathbb{T}^d)$ and orthogonal to $\varphi$ in the usual
$L^2$ sense. When $p\neq 2$, $\varphi$ is a Hilbert point in $H^p(\mathbb{T})$
if and only if $\varphi$ is a nonzero multiple of an inner function. An inner
function on $\mathbb{T}^d$ is a Hilbert point in any of the spaces
$H^p(\mathbb{T}^d)$, but there are other Hilbert points as well when $d\geq 2$.
We investigate the case of $1$-homogeneous polynomials in depth and obtain as a
byproduct a new proof of the sharp Khintchin inequality for Steinhaus variables
in the range $2<p<\infty$. We also study briefly the dynamics of a certain
nonlinear projection operator that characterizes Hilbert points as its fixed
points. We exhibit an example of a function $\varphi$ that is a Hilbert point
in $H^p(\mathbb{T}^3)$ for $p=2, 4$, but not for any other $p$; this is
verified rigorously for $p>4$ but only numerically for $1\leq p<4$.

10
The categories of this paper are: q-bio
11
The categories of this paper are: hep-lat
22
The categories of this paper are: chao-dyn
20
The categories of this paper are: acc-phys
13
The categories of this paper are: quant-ph
9
The categories of this paper are: nlin
17
The categories of this paper are: q-fin
25
The categories of this paper are: patt-sol
0
The categories of this paper are: hep-ph
7
The categories of this paper are: hep-th
33
The categories of this paper are: plasm-ph
21
The categories of this paper are: adap-org
5
The categories of this paper are: gr-qc
8
The categories of this paper are: hep-ex
6
The categories of this paper are: astro-ph
32
The categories of this paper are: atom-ph