Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 2 (2006), 033, 8 pages      math-ph/0603011      https://doi.org/10.3842/SIGMA.2006.033

A New Form of the Spherical Expansion of Zonal Functions and Fourier Transforms of SO(d)-Finite Functions

Agata Bezubik a and Aleksander Strasburger b
a) Institute of Mathematics, University of Bialystok, Akademicka 2, 15-267 Bialystok, Poland
b) Department of Econometrics and Informatics, Warsaw Agricultural University, Nowoursynowska 166, 02-787 Warszawa, Poland

Received November 30, 2005, in final form February 17, 2006; Published online March 03, 2006

Abstract
This paper presents recent results obtained by the authors (partly in collaboration with A. Dabrowska) concerning expansions of zonal functions on Euclidean spheres into spherical harmonics and some applications of such expansions for problems involving Fourier transforms of functions with rotational symmetry. The method used to derive the expansion formula is based entirely on differential methods and completely avoids the use of various integral identities commonly used in this context. Some new identities for the Fourier transform are derived and as a byproduct seemingly new recurrence relations for the classical Bessel functions are obtained.

Key words: spherical harmonics; zonal harmonic polynomials; Fourier-Laplace expansions; special orthogonal group; Bessel functions; Fourier transform; Bochner identity.

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