Local and global properties of functions and their Fourier transforms
type: Item
instance of: scholarly article
title: Local and global properties of functions and their Fourier transforms
zbMATH Open document ID: 0878.42006
author: John J. F. Fournier
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1998-01-08T00:00:00Z
review text: It is known that if an integrable function on the circle group \(\mathbb{T}\) with nonnegative Fourier coefficients is square-integrable in some neighbourhood of 0, then it is square-integrable on \(\mathbb{T}\) (by N. Wiener). However, an analogous result does not hold for the reals \(\mathbb{R}\) (by Kawazoe, Onoe and Tachizawa). The author investigates integrable functions with nonnegative Fourier transforms through the amalgam \((L^p,\ell^q)\), and he gives another proof for the latter. One of his results is as follows. Theorem: The following conditions are equivalent for any function in \(L^1(\mathbb{R})\) with a nonnegative transform. (a) The function is square-integrable in some neighbourhood of 0. (b) The function belongs to \((L^2,\ell^\infty)\). (c) The transform of the function belongs to \((L^1,\ell^2)\).
Mathematics Subject Classification ID: 42A38
Mathematics Subject Classification ID: 43A15
Mathematics Subject Classification ID: 43A25
Mathematics Subject Classification ID: 43A35
zbMATH DE Number: 1036424
zbMATH Keywords: integrable functions with nonnegative Fourier transforms
zbMATH Keywords: amalgam
reviewed by: Hiroshi Yamaguchi
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1036424