## smoothness classes in terms of decay of Fourier coefficients

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• Dear Colleagues, I have a question concerning smoothness classes defined in terms of decay of Fourier coefficients. I formulate my question as a conjecture. I
Message 1 of 2 , Aug 3, 2003
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Dear Colleagues,

I have a question concerning smoothness classes defined in terms
of decay of Fourier coefficients. I formulate my question as a
conjecture. I believe that the conjecture is true, but I was not able
neither to prove, nor to disprove it. I hope somebody will be able to do
this, or, at least, to give me a hint.

Best wishes, Alexei
------------------------------------------------------------------------------------
\documentclass[12pt]{article}
\newtheorem{conjecture}{Conjecture}
\newcommand{\ind}{\mathrm{ind}\,}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
Let ${\bf T}$ be the unit circle. For a complex-valued function $a\in L1({\bf T})$, let
$\{a_k\}_{k=-\infty}^\infty$ be the sequence of the Fourier coefficients
of $a$,
$a_k:=\frac{1}{2\pi}\int_0^{2\pi}a(e^{i\theta})e^{-ik\theta}d\theta.$
Let $W$ be the Wiener algebra of all functions $a$ on ${\bf T}$ of the form
$a(t)=\sum_{k=-\infty}^\infty a_k t^k \quad (t\in{\bf T})$
for which
$\|a\|_W:=\sum_{k=-\infty}^\infty |a_k|<\infty.$
It is well known that $W$ is a Banach algebra under the norm $\|\cdot\|_W$
and that $W$ is continuously imbedded into $C({\bf T})$, the Banach algebra
of all complex-valued continuous functions with the maximum norm.
We shall denote the Cauchy index of a continuous function $a$
by $\ind a$.

Let $p:[0,\infty)\to[0,\infty)$ be a right-continuous non-decreasing
function such that $p(0)=0$, $p(t)>0$ for $t>0$, and
$\lim\limits_{t\to\infty} p(t)=\infty$.
Then the function $q(s)=\sup\{t :\ p(t)\le s\}$
(defined for $s\ge 0$) has the same
properties as the function $p$. The convex functions $\Phi$ and $\Psi$
defined by the equalities
$\Phi(x):=\int_0^x p(t) dt, \quad \Psi(x):=\int_0^x q(s)ds \quad (x\ge 0)$
are called \textit{complementary $N$-functions}.
An $N$-function $\Phi$ is said to satisfy the $\Delta_20$-condition
if
$\limsup_{x\to 0}\frac{\Phi(2x)}{\Phi(x)}<\infty.$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{conjecture}
There exist sequences $\{\varphi_k\}_{k=0}^\infty, \{\psi_k\}_{k=0}^\infty$ of positvie numbers and constants
$C_\varphi,C_\psi,M\in(0,\infty)$ such that
\begin{enumerate}
\item[{\rm (a)}]
$\varphi_0=\psi_0=1$;
\item[{\rm (b)}]
for all $k\in{\bf N}$,
$\varphi_{k-1}\le\varphi_k, \quad \psi_{k-1}\le\psi_k; \quad \varphi_{2k}\le C_\varphi \varphi_k, \quad \psi_{2k}\le C_\psi\psi_k, \quad k\le M \varphi_k\psi_k;$
\end{enumerate}
there exist complementary $N$-functions $\Phi,\Psi$ both satisfying
the $\Delta_20$-condition, and there exists a function $a\in W$ such that
$a(t)\ne 0\ \mbox{for all}\ t\in{\bf T}, \quad\quad \ind a=0;$
for all $p\in(1,\infty)$ and all $\alpha\in[0,1]$,
$\sum_{k=1}^\infty \Big(|a_{-k}|(k+1)^\alpha\Big)^p + \sum_{k=0}^\infty \Big(|a_k|(k+1)^{1-\alpha}\Big)^{p/(p-1)}=\infty,$
but
$\sum_{k=1}^\infty \Phi(|a_{-k}|\varphi_k) + \sum_{k=0}^\infty \Psi(|a_k|\psi_k)<\infty.$
\end{conjecture}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}

--
Alexei Yu. Karlovich (Oleksiy Karlovych)
Departamento de Matematica
Instituto Superior Tecnico
Av. Rovisco Pais
1049-001 Lisboa,
Portugal

Phone: +351-21 8417037
Fax: +351-21 8417598
E-mail: akarlov@...
http://www.math.ist.utl.pt/~akarlov
• Dear Colleagues, I together with Pedro Santos proved the conjecture constructively with the simplest $N$-functions $Phi(x)= Psi(x)=x^2$ and specially chosen
Message 2 of 2 , Aug 8, 2003
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Dear Colleagues,

I together with Pedro Santos proved the conjecture
constructively with the simplest $N$-functions
$\Phi(x)=\Psi(x)=x^2$ and specially chosen weight
sequences. For the proof, see Lemma 1.3 and
Section 6.3 in our preprint

A. Karlovich, P. Santos,
On asymptotics of Toeplitz determinants with symbols
of nonstandard smoothness.

This paper is available on my web page

http://www.math.ist.utl.pt/~akarlov/submitted.html

Best wishes, Alexei

Alexei Karlovich wrote:
> Dear Colleagues,
>
> I have a question concerning smoothness classes defined in terms
> of decay of Fourier coefficients. I formulate my question as a
> conjecture. I believe that the conjecture is true, but I was not able
> neither to prove, nor to disprove it. I hope somebody will be able to do
> this, or, at least, to give me a hint.
>
> Best wishes, Alexei
> ------------------------------------------------------------------------------------
> \documentclass[12pt]{article}
> \newtheorem{conjecture}{Conjecture}
> \newcommand{\ind}{\mathrm{ind}\,}
> %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
> \begin{document}
> Let ${\bf T}$ be the unit circle. For a complex-valued function $a\in > L1({\bf T})$, let
> $\{a_k\}_{k=-\infty}^\infty$ be the sequence of the Fourier coefficients
> of $a$,
> $> a_k:=\frac{1}{2\pi}\int_0^{2\pi}a(e^{i\theta})e^{-ik\theta}d\theta. >$
> Let $W$ be the Wiener algebra of all functions $a$ on ${\bf T}$ of the form
> $> a(t)=\sum_{k=-\infty}^\infty a_k t^k > \quad (t\in{\bf T}) >$
> for which
> $> \|a\|_W:=\sum_{k=-\infty}^\infty |a_k|<\infty. >$
> It is well known that $W$ is a Banach algebra under the norm $\|\cdot\|_W$
> and that $W$ is continuously imbedded into $C({\bf T})$, the Banach algebra
> of all complex-valued continuous functions with the maximum norm.
> We shall denote the Cauchy index of a continuous function $a$
> by $\ind a$.
>
> Let $p:[0,\infty)\to[0,\infty)$ be a right-continuous non-decreasing
> function such that $p(0)=0$, $p(t)>0$ for $t>0$, and
> $\lim\limits_{t\to\infty} p(t)=\infty$.
> Then the function $q(s)=\sup\{t :\ p(t)\le s\}$
> (defined for $s\ge 0$) has the same
> properties as the function $p$. The convex functions $\Phi$ and $\Psi$
> defined by the equalities
> $> \Phi(x):=\int_0^x p(t) dt, > \quad > \Psi(x):=\int_0^x q(s)ds > \quad (x\ge 0) >$
> are called \textit{complementary $N$-functions}.
> An $N$-function $\Phi$ is said to satisfy the $\Delta_20$-condition
> if
> $> \limsup_{x\to 0}\frac{\Phi(2x)}{\Phi(x)}<\infty. >$
> %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
> \begin{conjecture}
> There exist sequences $\{\varphi_k\}_{k=0}^\infty, > \{\psi_k\}_{k=0}^\infty$ of positvie numbers and constants
> $C_\varphi,C_\psi,M\in(0,\infty)$ such that
> \begin{enumerate}
> \item[{\rm (a)}]
> $\varphi_0=\psi_0=1$;
> \item[{\rm (b)}]
> for all $k\in{\bf N}$,
> $> \varphi_{k-1}\le\varphi_k, > \quad > \psi_{k-1}\le\psi_k; > \quad > \varphi_{2k}\le C_\varphi \varphi_k, > \quad > \psi_{2k}\le C_\psi\psi_k, > \quad > k\le M \varphi_k\psi_k; >$
> \end{enumerate}
> there exist complementary $N$-functions $\Phi,\Psi$ both satisfying
> the $\Delta_20$-condition, and there exists a function $a\in W$ such that
> $> a(t)\ne 0\ \mbox{for all}\ t\in{\bf T}, > \quad\quad > \ind a=0; >$
> for all $p\in(1,\infty)$ and all $\alpha\in[0,1]$,
> $> \sum_{k=1}^\infty \Big(|a_{-k}|(k+1)^\alpha\Big)^p > + > \sum_{k=0}^\infty \Big(|a_k|(k+1)^{1-\alpha}\Big)^{p/(p-1)}=\infty, >$
> but
> $> \sum_{k=1}^\infty \Phi(|a_{-k}|\varphi_k) > + > \sum_{k=0}^\infty \Psi(|a_k|\psi_k)<\infty. >$
> \end{conjecture}
> %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
> \end{document}
>

--
Alexei Yu. Karlovich (Oleksiy Karlovych)
Departamento de Matematica
Instituto Superior Tecnico
Av. Rovisco Pais
1049-001 Lisboa,
Portugal

Phone: +351-21 8417037
Fax: +351-21 8417598
E-mail: akarlov@...
http://www.math.ist.utl.pt/~akarlov
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