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

S. J. Bhatt, P. A. Dabhi, and H. V. Dedania. "Beurling algebra analogues of theorems of Wiener-Lévy-Żelazko and Żelazko." Studia Mathematica 195.3 (2009): 219-225. <http://eudml.org/doc/284428>.

@article{S2009,
abstract = {Let 0 < p ≤ 1, let ω: ℤ → [1,∞) be a weight on ℤ and let f be a nowhere vanishing continuous function on the unit circle Γ whose Fourier series satisfies $∑_\{n∈ℤ\} |f̂(n)|^\{p\}ω(n) < ∞$. Then there exists a weight ν on ℤ such that $∑_\{n∈ℤ\} |\widehat\{(1/f)\}(n)|^\{p\} ν(n) < ∞$. Further, ν is non-constant if and only if ω is non-constant; and ν = ω if ω is non-quasianalytic. This includes the classical Wiener theorem (p = 1, ω = 1), Domar theorem (p = 1, ω is non-quasianalytic), Żelazko theorem (ω = 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 ω and 0 < p < 1, the locally bounded space $L^\{p\}(G,ω)$ 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.},
author = {S. J. Bhatt, P. A. Dabhi, H. V. Dedania},
journal = {Studia Mathematica},
language = {eng},
number = {3},
pages = {219-225},
title = {Beurling algebra analogues of theorems of Wiener-Lévy-Żelazko and Żelazko},
url = {http://eudml.org/doc/284428},
volume = {195},
year = {2009},
}

TY - JOUR
AU - S. J. Bhatt
AU - P. A. Dabhi
AU - H. V. Dedania
TI - Beurling algebra analogues of theorems of Wiener-Lévy-Żelazko and Żelazko
JO - Studia Mathematica
PY - 2009
VL - 195
IS - 3
SP - 219
EP - 225
AB - Let 0 < p ≤ 1, let ω: ℤ → [1,∞) be a weight on ℤ and let f be a nowhere vanishing continuous function on the unit circle Γ whose Fourier series satisfies $∑_{n∈ℤ} |f̂(n)|^{p}ω(n) < ∞$. Then there exists a weight ν on ℤ such that $∑_{n∈ℤ} |\widehat{(1/f)}(n)|^{p} ν(n) < ∞$. Further, ν is non-constant if and only if ω is non-constant; and ν = ω if ω is non-quasianalytic. This includes the classical Wiener theorem (p = 1, ω = 1), Domar theorem (p = 1, ω is non-quasianalytic), Żelazko theorem (ω = 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 ω and 0 < p < 1, the locally bounded space $L^{p}(G,ω)$ 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.
LA - eng
UR - http://eudml.org/doc/284428
ER -