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

• Autorzy: S. J. Bhatt , P. A. Dabhi , H. V. Dedania
• Języki publikacji: EN

Abstrakty

EN

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.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2009
• Tom: 195
• Numer: 3
• Strony: 219-225

Daty

wydano

2009

Twórcy

autor

S. J. Bhatt

• Department of Mathematics, Sardar Patel University, Vallabh Vidyanagar-388 120, Gujarat, India

autor

P. A. Dabhi

• Department of Mathematics, Sardar Patel University, Vallabh Vidyanagar-388 120, Gujarat, India

autor

H. V. Dedania

• Department of Mathematics, Sardar Patel University, Vallabh Vidyanagar-388 120, Gujarat, India

Identyfikatory

DOI

10.4064/sm195-3-2