On spherical expansions of zonal functions on Euclidean spheres |
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Authors: | Agata Bezubik Agata Da̧browska Aleksander Strasburger |
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Affiliation: | (1) Institute of Mathematics, University of Białystok, Akademicka 2, 15-267 Białystok, Poland;(2) Department of Econometrics and Statistics, Warsaw University of Life Science (SGGW), Nowoursynowska 166, 02-787 Warszawa, Poland |
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Abstract: | This paper describes a new approach to the problem of computing spherical expansions of zonal functions on Euclidean spheres. We derive an explicit formula for the coefficients of the expansion expressing them in terms of the Taylor coefficients of the profile function rather than (as done usually) in terms of its integrals against Gegenbauer polynomials. Our proof of this result is based on a polynomial identity equivalent to the canonical decomposition of homogeneous polynomials and uses only basic properties of this decomposition together with simple facts concerning zonal harmonic polynomials. As corollaries, we obtain direct and apparently new derivations of the so-called plane wave expansion and of the expansion of the Poisson kernel for the unit ball. Received: 26 January 2007 |
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Keywords: | Primary 33C55, 43A85 Secondary 33C80, 33C45 |
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