Local harmonic analysis on spheres
type: Item
instance of: scholarly article
title: Local harmonic analysis on spheres
zbMATH Open document ID: 0651.43006
DOI: 10.1016/0022-1236(88)90095-X
author: Robert S. Strichartz
published in: Journal of Functional Analysis
publication date: 1988-01-01T00:00:00Z
review text: A continuous expansion of the form \(f(x)=\int^{\infty}_{0}f_{\lambda}(x)h(\lambda)d\lambda\) is established for functions or distributions supported on a compact set K that is small in the sense that no pair of antipodal points can both belong to K in the sphere S n, valid in a neighbourhood \(\Omega\) of K consisting of the complement of the antipodal points of K, where \(f_{\lambda}(x)\) are eigenfunctions of the Laplacian of \(\Omega\) with eigenvalues ((n-1)/2) 2-\(\lambda\) 2 and h(\(\lambda)\) can be given explicitly as follows. When n is odd, \[ h(\lambda)=1/(2\pi)^{\sigma +1}\gamma_ n\prod^{\sigma -1}_{k=0}(\lambda \quad 2-k\quad 2),\quad \gamma_ n=(n-2)(n-4)...1 \] and when n is even \[ h(\lambda)=\lambda /(2\pi)^{\sigma +1/2}\gamma_ n\prod^{\sigma -3/2}_{k=0}(\lambda \quad 2-(k+1/2)\quad 2)h_ 1(\lambda),\quad \gamma_ n=(n-2)(n-4)...2 \] where \(h_ 1(\lambda)=1/\pi \int^{\pi}_{0}(\sin \lambda_ s/\sin s/2)ds\), for \(\sigma =(n-1)/2\) and \(\lambda =\sigma +k\), \(k=0,1,2,...\). The function \(f_{\lambda}(x)\) can be given by \(f_{\lambda}(x)=\int_{K}\phi_{\lambda}(d(x,y))f(y)dy\) where d(x,y) denotes the distance on the sphere S n and \(\phi_{\lambda}\) are certain zonal functions. When n is odd, the function \(f_{\lambda}(x)\) can be uniquely characterized by analyticity and growth conditions in \(\lambda\) of Paley-Wiener type. The expansion is applied to study functions of the Laplacian and operators that commute with the Laplacian.
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 43A90
Mathematics Subject Classification ID: 41A58
zbMATH DE Number: 4061876
zbMATH Keywords: spheres
zbMATH Keywords: spherical harmonics
zbMATH Keywords: distributions
zbMATH Keywords: antipodal points
zbMATH Keywords: eigenfunctions of the Laplacian
zbMATH Keywords: zonal functions
zbMATH Keywords: analyticity
zbMATH Keywords: growth conditions
zbMATH Keywords: Paley-Wiener type
reviewed by: Koichi Saka
MaRDI profile type: Publication
cites work: Q5847356
cites work: Q5724171
cites work: A duality for symmetric spaces with applications to group representations
cites work: Q3215519
cites work: Conformally invariant differential equations and projective geometry
cites work: Hilbert and Fourier Transforms on a Sphere
cites work: Fourier Analysis on the Sphere
cites work: Q5640160
cites work: Mean Value Properties of the Laplacian via Spectral Theory
cites work: Sur les représentations unitaires des groupes de Lorentz généralisés
cites work: A symbol calculus for a class of pseudodifferential operators on \(S^ n\) and band asymptotics
cites work: Q5654049
cites work: Q3872897
cites work: A Proof of a Formula in Fourier Analysis on the Sphere
cites work: Pseudo-differential analysis on hyperbolic surfaces
description: scientific article; zbMATH DE number 4061876