- 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 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