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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {S}^d \cong SO(d+1)/SO(d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mi>d</mi> </msup> <mo>≅</mo> <mi>S</mi> <mi>O</mi> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <mi>S</mi> <mi>O</mi> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {R}^{d+1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>d</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>. We exploit a correspondence between G-invariant kernels on <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$G/K \times G/K$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo stretchy="false">/</mo> <mi>K</mi> <mo>×</mo> <mi>G</mi> <mo stretchy="false">/</mo> <mi>K</mi> </mrow> </math></EquationSource> </InlineEquation> and K-bi-invariant functions on G to show that the Delsarte-type problem on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$G/K \times G/K$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo stretchy="false">/</mo> <mi>K</mi> <mo>×</mo> <mi>G</mi> <mo stretchy="false">/</mo> <mi>K</mi> </mrow> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$G/K \times G/K$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo stretchy="false">/</mo> <mi>K</mi> <mo>×</mo> <mi>G</mi> <mo stretchy="false">/</mo> <mi>K</mi> </mrow> </math></EquationSource> </InlineEquation>. 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.

Delsarte-Type Extremal Problems and Convolution Roots on Homogeneous Spaces

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2026年4月7日

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Mita Ramabulana

摘要

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 $\mathbb {S}^d \cong SO(d+1)/SO(d)$ $S^{d} ≅ S O (d + 1) / S O (d)$ in $\mathbb {R}^{d+1}$ $R^{d + 1}$ . We exploit a correspondence between G-invariant kernels on $G/K \times G/K$ $G / K \times G / K$ and K-bi-invariant functions on G to show that the Delsarte-type problem on $G/K \times G/K$ $G / K \times G / K$ 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 $G/K \times G/K$ $G / K \times G / K$ . 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.