A spherical Wiener-Plancherel formula
type: Item
instance of: scholarly article
title: A spherical Wiener-Plancherel formula
zbMATH Open document ID: 0767.42004
DOI: 10.1016/0022-247X(92)90355-H
author: George Benke
published in: Journal of Mathematical Analysis and Applications
publication date: 1993-04-01T00:00:00Z
review text: Wiener developed a modification of the Fourier transform in one dimension, \(s(\gamma)\), that converges for functions whose square integral can diverge linearly, \[ \sup_{T>0} \frac 1 T \int_{- T}^{T} | f(t)| ^ 2 dt < +\infty. \] The formula for \(s\) is \[ s(\gamma) = \int_{| t| <1} f(t) \frac{e^{-i \gamma t} - 1}{i t} dt + \int_{| t| >1} f(t) \frac{e^{-i \gamma t}}{-i t} dt, \] and the generalization of the Plancherel formula is: \[ \lim_{T \rightarrow \infty} \frac 1 T \int_{-T}^ T | f(t)| ^ 2 dt = \lim_{\epsilon \rightarrow 0} \frac 1{4\pi \epsilon}\int_{- \infty}^{\infty} | s(\gamma + \epsilon)- s(\gamma - \epsilon)| ^ 2 d\gamma, \] where this means that if either limit exists, the other exists and they agree. In \(d\)-dimensional space, a Wiener-Plancherel formula takes the form \[ \lim_{T \rightarrow \infty} \frac 1{| A_ T| } \int_{A_ T} | f(t)| ^ 2 dt = \lim_{\epsilon \rightarrow 0} \frac{\nu}{\phi(\epsilon)} \int_{R^ d} | \nabla_{\epsilon} s(\gamma)| ^ 2 d\gamma, \] where \(\{A_ T\}\) is some increasing family of sets whose union is \(R^ d\), \(| A_ T| \) is the Lebesgue measure of \(A_ T\), \(\phi(\epsilon)\) is a function which converges to \(0\) as \(\epsilon\) converges to \(0\), \(\nu\) is a normalizing constant, \(s\) is a formal solution of \(Ls = \widehat f\), \(L\) is a differential operator and \(\nabla_{\epsilon}\) is a difference operator associated with \(L\). In \textit{J. Benedetto, G. Benke} and \textit{W. Evans} [Adv. Appl. Math. 10, No. 4, 457-487 (1989; Zbl 0703.42008)] this was proved when the sets \(\{A_ T\}\) are rectangles, \(L = \partial^ n/\partial \gamma_ 1 \dots \partial \gamma_ n\), and \(\nabla_{\epsilon}\) is a discrete symmetric difference operator. The convergence criterion is roughly that the limit exists for every increasing sequence of rectangles of fixed shape and all the limits are the same. \textit{J. Benedetto} [SIAM J. Math. Anal. 22, No. 4, 1110-1130 (1991; Zbl 0731.42015)] has considered a spherical case where \(L\…
Mathematics Subject Classification ID: 42B10
zbMATH DE Number: 150704
zbMATH Keywords: spherical Wiener-Plancherel formula
zbMATH Keywords: modification of the Fourier transform
reviewed by: Raymond Johnson
MaRDI profile type: Publication
cites work: The Spherical Wiener–Plancherel Formula and Spectral Estimation
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description: scientific article; zbMATH DE number 150704