Delsarte-Type Extremal Problems and Convolution Roots on Homogeneous Spaces

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 Rᵈ⁺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.