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 $$\mathbb {S}^d \cong SO(d+1)/SO(d)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>S</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo>≅</mml:mo> <mml:mi>S</mml:mi> <mml:mi>O</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>/</mml:mo> <mml:mi>S</mml:mi> <mml:mi>O</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> in $$\mathbb {R}^{d+1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> . We exploit a correspondence between G -invariant kernels on $$G/K \times G/K$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>/</mml:mo> <mml:mi>K</mml:mi> <mml:mo>×</mml:mo> <mml:mi>G</mml:mi> <mml:mo>/</mml:mo> <mml:mi>K</mml:mi> </mml:mrow> </mml:math> and K -bi-invariant functions on G to show that the Delsarte-type problem on $$G/K \times G/K$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>/</mml:mo> <mml:mi>K</mml:mi> <mml:mo>×</mml:mo> <mml:mi>G</mml:mi> <mml:mo>/</mml:mo> <mml:mi>K</mml:mi> </mml:mrow> </mml:math> 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 $$G/K \times G/K$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>/</mml:mo> <mml:mi>K</mml:mi> <mml:mo>×</mml:mo> <mml:mi>G</mml:mi> <mml:mo>/</mml:mo> <mml:mi>K</mml:mi> </mml:mrow> </mml:math> . 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.