Computing Fourier transforms and convolutions on the 2-sphere
type: Item
instance of: scholarly article
title: Computing Fourier transforms and convolutions on the 2-sphere
zbMATH Open document ID: 0801.65141
author: James R. Driscoll
author: Dennis Healy
published in: Advances in Applied Mathematics
publication date: 1994-08-10T00:00:00Z
review text: This important paper deals with the efficient computation of the spherical harmonic expansion, or Fourier transform, of functions defined on the unit sphere \(S^ 2\). For example, the Fourier transform on \(S^ 2\) appears in tomography, geophysics, seismology, atmospheric science, and crystallography. The known Fourier analysis on \(S^ 2\) is used to prove a sampling theorem for band-limited functions. This sampling theorem allows the exact computation of the Fourier transforms of a band- limited function as finite sums of sampled data at an equiangular grid. In order to construct an efficient scheme for evaluating these sums, the authors describe an \(O(n( \log n)^ 2)\) time algorithm for the Legendre polynomial transform of a function sampled at the \(n = 2^ k\) nodes \(\cos (2 \pi j/n)\) \((j = 0, \dots, n - 1)\). Theoretical and experimental results concerning the effect of finite precision arithmetic on this algorithm are described, too. Finally, the procedure is extended to the associated Legendre functions to obtain a fast algorithm of the Fourier transforms on \(S^ 2\).
reviewed by: Manfred Tasche
Mathematics Subject Classification ID: 65T40
Mathematics Subject Classification ID: 43A25
Mathematics Subject Classification ID: 42B10
zbMATH DE Number: 617366
zbMATH Keywords: convolutions
zbMATH Keywords: Fourier transform on the sphere
zbMATH Keywords: numerical stability
zbMATH Keywords: spherical harmonic expansion
zbMATH Keywords: tomography
zbMATH Keywords: geophysics
zbMATH Keywords: seismology
zbMATH Keywords: atmospheric science
zbMATH Keywords: crystallography
zbMATH Keywords: sampling theorem
zbMATH Keywords: band-limited functions
zbMATH Keywords: Legendre polynomial transform
zbMATH Keywords: fast algorithm
MaRDI profile type: Publication
full work available at URL: aama.1994.1008
OpenAlex ID: W2160615084
DOI: 10.1006/AAMA.1994.1008
description: scientific article; zbMATH DE number 617366