Abstract: We start with a brief introduction to asymptotic harmonic analysis and Olshanski spherical pairs, which are limits of increasing sequences of Gelfand pairs (G,K) (e.g. the rank of G/K grows). For certain classes of Olshanski pairs we describe how spherical functions can be approximated by those of the underlying Gelfand pairs. This asymptotic behaviour of spherical functions (e.g. for large rank) turns out to be a special case of more general asymptotics for special functions from Dunkl theory.
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