Institute of Mathematics,

University of Bialystok,

2 Akademicka Str.,

15-267 Bialystok,

POLAND

E-mail: agatab@math.uwb.edu.pl

**On spherical expansions of zonal functions and Fourier
transforms of SO(d)-finite measures**

**Abstract:**

The article presents a simple new approach to the determination of
the spherical expansion of zonal functions on euclidean space. We use the
standard representation theoretic properties of spherical harmonics and
the explicit form of the reproducing kernels for these spaces by means
of classical Gegenbauer polynomials. As a special case we reobtain by this
method the so called plane wave expansion and the expansion of the Poisson
kernel. We present a new method of computing the Fourier transform of SO(d)-finite
functions on the unit sphere which enables us to reobtain the classical
Bochner identity.