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

Part of: Abstract harmonic analysis Harmonic analysis in one variable

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: menegatt@gmail.com
C. P. Oliveira
Affiliation:
Instituto de Matemática e Computação, Universidade Federal de Itajubá, 37500-903 Itajubá MG, Brazil Email: oliveira@unifei.edu.br
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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.

References

Barbosa, V. S. and Menegatto, V. A., Strictly positive definite kernels on compact two-point homogeneous spaces. Math. Inequal. Appl. 19(2016), 743–756. https://doi.org/10.7153/mia-19-54Google Scholar
Berg, C., Christensen, J. P. R., and Ressel, P., Harmonic analysis on semigroups. Theory of positive definite and related functions. Graduate Texts in Mathematics, 100, Springer-Verlag, New York, 1984. https://doi.org/10.1007/978-1-4612-1128-0CrossRefGoogle Scholar
Berg, C. and Porcu, E., From Schoenberg coefficients to Schoenberg functions. Constr. Approx. 45(2017), 217–241. https://doi.org/10.1007/s00365-016-9323-9CrossRefGoogle Scholar
Chen, D., Menegatto, V. A., and Sun, X., A necessary and sufficient condition for strictly positive definite functions on spheres. Proc. Amer. Math. Soc. 131(2003), 2733–2740. https://doi.org/10.1090/S0002-9939-03-06730-3CrossRefGoogle Scholar
Estrade, A., Fariñas, A., and Porcu, E., Characterization theorems for covariance functions on the $n$-dimensional sphere across time. hal-01417668v2, 2017.Google Scholar
Estrade, A., Fariñas, A., and Porcu, E., Covariance functions on spheres cross time: beyond spatial isotropy and temporal stationarity. Statist. Probab. Lett. 151(2019), 1–7. https://doi.org/10.1016/j.spl.2019.03.011CrossRefGoogle Scholar
Folland, G. B., A course in abstract harmonic analysis. Second ed., Textbooks in Mathematics, CRC Press, Boca Raton, 2016.CrossRefGoogle Scholar
Gangolli, R., Positive definite kernels on homogeneous spaces and certain stochastic processes related to Lévy’s Brownian motion of several parameters. Ann. Inst. H. Poincaré Sect. B (N.S.) 3(1967), 121–226.Google Scholar
Guella, J. C. and Menegatto, V. A., Schoenberg’s theorem for positive definite functions on products: a unifying framework. J. Fourier Anal. Appl. 25(2019), 1424–1446. https://doi.org/10.1007/s00041-018-9631-5CrossRefGoogle Scholar
Rudin, W., Fourier analysis on groups. Reprint of the 1962 original. Wiley Classics Library. A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1990. https://doi.org/10.1002/9781118165621CrossRefGoogle Scholar
Schoenberg, I. J., Positive definite functions on spheres. Duke Math. J. 9(1942), 96–108.CrossRefGoogle Scholar
Wang, H.-S., Two-point homogeneous spaces. Ann. Math. 55(1952), 177–191. https://doi.org/10.2307/1969427CrossRefGoogle Scholar
Wendland, H., Scattered data approximation. Cambridge Monographs on Applied and Computational Mathematics, 17, Cambridge University Press, Cambridge, 2005.Google Scholar
White, P. and Porcu, E., Towards a complete picture of stationary covariance functions on spheres cross time. Electron. J. Stat. 13(2019), 2566–2594. https://doi.org/10.1214/19-EJS1593CrossRefGoogle Scholar