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## Studia Mathematica

2009 | 195 | 3 | 219-225
Tytuł artykułu

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

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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.
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Tom
Numer
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219-225
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wydano
2009
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autor
• Department of Mathematics, Sardar Patel University, Vallabh Vidyanagar-388 120, Gujarat, India
autor
• Department of Mathematics, Sardar Patel University, Vallabh Vidyanagar-388 120, Gujarat, India
autor
• Department of Mathematics, Sardar Patel University, Vallabh Vidyanagar-388 120, Gujarat, India
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