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Positive Definiteness on Products of Compact Two-point Homogeneous Spaces and Locally Compact Abelian Groups

Part of: Harmonic analysis in one variable Abstract harmonic analysis

Published online by Cambridge University Press:  29 October 2019

V. A. Menegatto  and
C. P. Oliveira
V. A. Menegatto
Affiliation:
Departamento de Matemática, ICMC-USP - São Carlos, Caixa Postal 668, 13560-970 São Carlos SP, Brazil Email: [email protected]
C. P. Oliveira
Affiliation:
Instituto de Matemática e Computação, Universidade Federal de Itajubá, 37500-903 Itajubá MG, Brazil Email: [email protected]
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Abstract

In this paper, we consider the problem of characterizing positive definite functions on compact two-point homogeneous spaces cross locally compact abelian groups. For a locally compact abelian group $G$ with dual group $\widehat{G}$, a compact two-point homogeneous space $\mathbb{H}$ with normalized geodesic distance $\unicode[STIX]{x1D6FF}$ and a profile function $\unicode[STIX]{x1D719}:[-1,1]\times G\rightarrow \mathbb{C}$ satisfying certain continuity and integrability assumptions, we show that the positive definiteness of the kernel $((x,u),(y,v))\in (\mathbb{H}\times G)^{2}\mapsto \unicode[STIX]{x1D719}(\cos \unicode[STIX]{x1D6FF}(x,y),uv^{-1})$ is equivalent to the positive definiteness of the Fourier transformed kernels $(x,y)\in \mathbb{H}^{2}\mapsto \widehat{\unicode[STIX]{x1D719}}_{\cos \unicode[STIX]{x1D6FF}(x,y)}(\unicode[STIX]{x1D6FE})$, $\unicode[STIX]{x1D6FE}\in \widehat{G}$, where $\unicode[STIX]{x1D719}_{t}(u)=\unicode[STIX]{x1D719}(t,u)$, $u\in G$. We also provide some results on the strict positive definiteness of the kernel.

Keywords

compact two-point homogeneous spacelocally compact abelian grouppositive definitenessFourier transformdual groupFourier inversion

MSC classification

Primary: 42A16: Fourier coefficients, Fourier series of functions with special properties, special Fourier series
Secondary: 42A82: Positive definite functions 43A35: Positive definite functions on groups, semigroups, etc. 43A85: Analysis on homogeneous spaces
Type
Article
Information
Canadian Mathematical Bulletin , Volume 63 , Issue 4 , December 2020 , pp. 705 - 715
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Copyright
© Canadian Mathematical Society 2019

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Footnotes

Partially supported by FAPESP, grant 2016/09906-0.

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