A New Form of the Spherical Expansion of Zonal Functions and Fourier Transforms of $\mathrm{SO}(d)$-Finite Functions

This paper presents recent results obtained by the authors (partly in collaboration with A. Da̧browska) 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.