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Beurling algebra analogues of theorems of Wiener–Lévy–Żelazko and Żelazko

Tom 195 / 2009

S. J. Bhatt, P. A. Dabhi, H. V. Dedania Studia Mathematica 195 (2009), 219-225 MSC: 42A16, 42A28, 46J05. DOI: 10.4064/sm195-3-2

Streszczenie

Let $0< p\leq 1$, let $\omega :\mathbb Z\rightarrow [1,\infty)$ be a weight on $\mathbb{Z}$ and let $f$ be a nowhere vanishing continuous function on the unit circle $\mit\Gamma$ whose Fourier series satisfies $\sum _{n\in \mathbb{Z}}{|\widehat f(n)|^p \omega(n)} < \infty$. Then there exists a weight $\nu$ on $\mathbb{Z}$ such that $\sum _{n\in \mathbb{Z}}{|\widehat{(1/f)}(n)|^p \nu(n)} < \infty$. Further, $\nu$ is non-constant if and only if $\omega$ is non-constant; and $\nu=\omega$ if $\omega$ is non-quasianalytic. This includes the classical Wiener theorem ($p=1$, $\omega= 1$), Domar theorem ($p=1$, $\omega$ is non-quasianalytic), Żelazko theorem ($\omega=1$) and a recent result of Bhatt and Dedania ($p=1$). An analogue of the Lévy theorem at the present level of generality is also developed. Given a locally compact group $G$ with a continuous weight $\omega$ and $0 < p < 1$, the locally bounded space $L^p(G,\omega)$ is closed under convolution if and only if $G$ is discrete if and only if $G$ admits an atom. This generalizes and refines another result of Żelazko.

Autorzy

  • S. J. BhattDepartment of Mathematics
    Sardar Patel University
    Vallabh Vidyanagar-388 120
    Gujarat, India
    e-mail
  • P. A. DabhiDepartment of Mathematics
    Sardar Patel University
    Vallabh Vidyanagar-388 120
    Gujarat, India
    e-mail
  • H. V. DedaniaDepartment of Mathematics
    Sardar Patel University
    Vallabh Vidyanagar-388 120
    Gujarat, India
    e-mail

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