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smoothness classes in terms of decay of Fourier coefficients

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  • Alexei Karlovich
    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
    • 0 Attachment
      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
    • Alexei Karlovich
      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
      • 0 Attachment
        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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