Delsarte-Type Extremal Problems and Convolution Roots on Homogeneous Spaces

Abstract For a locally compact group G and compact subgroup K, we consider a Delsarte-type extremal problem for G -invariant positive definite kernels on the homogeneous space G / K, generalising a certain Turán problem for isotropic positive definite kernels on the unit sphere Sᵈ SO (d+1) /SO (d) S d ≅ S O (d + 1) / S O (d) in R^d+1 R d + 1. We exploit a correspondence between G -invariant kernels on G/K G/K G / K × G / K and K -bi-invariant functions on G to show that the Delsarte-type problem on G/K G/K G / K × G / K is equivalent to a Delsarte-type problem for K -bi-invariant functions on G. We use this correspondence to show the existence of an extremal function for the Delsarte problem on G/K G/K G / K × G / K. In the case where (G, K) is a compact Gelfand pair, we show the existence of K -bi-invariant convolution roots for positive definite K -bi-invariant functions, consequently obtaining the existence of a G -invariant convolution root for G -invariant positive definite kernels.