Fourier transforms of Lipschitz functions on the hyperbolic plane \(H^2\)
type: Item
instance of: scholarly article
title: Fourier transforms of Lipschitz functions on the hyperbolic plane \(H^2\)
zbMATH Open document ID: 0903.42002
DOI: 10.1155/S0161171298000544
author: M. S. Younis
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 1999-01-05T00:00:00Z
full work available at URL: https://eudml.org/doc/48035
review text: The purpose of the present work is to study the order of magnitude of the Fourier transforms \(\widehat f(\lambda)\) for large \(\lambda\) of complex-valued functions \(f(z)\) satisfying certain Lipschitz conditions in the non-Euclidean hyperbolic plane \(H^2\). The main result is stated as Theorem. Let \(f(z)\) be a complex-valued function on the unit disc \(D\) and \(B=\partial D\). Then the conditions \[ \| f(z+ h)- f(z)\|_2= O(| h|^\alpha)\quad\text{as }| h|\to 0,\;0<\alpha< 1, \] and \[ \int^\infty_\lambda \int_B|\widehat f|^2 d\lambda db= O(\lambda^{-2\alpha- 1})\quad\text{as }\lambda\to \infty \] are equivalent.
Mathematics Subject Classification ID: 42B10
zbMATH DE Number: 1140285
zbMATH Keywords: spherical transforms
zbMATH Keywords: Fourier transforms
zbMATH Keywords: Lipschitz conditions
MaRDI profile type: Publication
OpenAlex ID: W2113593537
description: scientific article; zbMATH DE number 1140285