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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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