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(Strict) positive definite functions and differentiability

Abstract

The aim of this project is to obtain new results related to two lines of research: differentiability of positive definite functions (PD) and characterization of strictly positive definite functions (SPD). Recently, Buescu and Paixão ([BP11]) proved that given a PD function in $\mathbb{R}$, if some even order derivative in the origin is zero, then the function is constant, and they also obtained sufficient conditions on the even order derivatives in the origin for the function to be analytic, producing the maximal stripe where it can be holomorphically extended. In [BP14], analogous results were obtained for the case of PD functions in $\mathbb{C}$.Our purpose is to extend the results of Buescu and Paixão from [BP11, BP14] to the cases $\mathbb{R}^m$ and $\mathbb{C}^m$, $m\geq2$. About our second line of research, Chen, Menegatto and Sun ([CMS03]) obtained a full characterization of the SPD functions in $S^{m-1}$, $m\geq2$, in the classical case (scalar functions). Here we propose to obtain a characterization for the PD and SPD functions, in different settings. (AU)

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Scientific publications (8)
(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)
BERG, CHRISTIAN; PERON, ANA P.; PORCU, EMILIO. Orthogonal expansions related to compact Gelfand pairs. EXPOSITIONES MATHEMATICAE, v. 36, n. 3-4, SI, p. 259-277, . (16/03015-7, 14/25796-5)
MASSA, EUGENIO; PERON, ANA PAULA; PORCU, EMILIO. Positive Definite Functions on Complex Spheres and their Walks through Dimensions. Symmetry Integrability and Geometry-Methods and Applications, v. 13, . (14/25398-0, 16/03015-7, 14/25796-5)
GUELLA, J. C.; MENEGATTO, V. A.; PERON, A. P.. AN EXTENSION OF A THEOREM OF SCHOENBERG TO PRODUCTS OF SPHERES. Banach Journal of Mathematical Analysis, v. 10, n. 4, p. 671-685, . (12/22161-3, 14/25796-5, 14/00277-5)
CASTRO, MARIO H.; MASSA, EUGENIO; PERON, ANA PAULA. Characterization of Strict Positive Definiteness on products of complex spheres. POSITIVITY, v. 23, n. 4, p. 853-874, . (16/03015-7, 14/25796-5, 14/25398-0, 16/09906-0)
BERG, CHRISTIAN; PERON, ANA P.; PORCU, EMILIO. Schoenberg's theorem for real and complex Hilbert spheres revisited. Journal of Approximation Theory, v. 228, p. 58-78, . (16/03015-7, 14/25796-5, 16/09906-0)
GUELLA, JEAN C.; MENEGATTO, VALDIR A.; PERON, ARIA P.. Strictly Positive Definite Kernels on a Product of Spheres II. Symmetry Integrability and Geometry-Methods and Applications, v. 12, . (16/03015-7, 14/25796-5, 14/00277-5)
MASSA, E.; PERON, A. P.; PIANTELLA, A. C.. Estimates on the derivatives and analyticity of positive definite functions on R-m. ANALYSIS MATHEMATICA, v. 43, n. 1, p. 89-98, . (14/25796-5, 14/25398-0)
GUELLA, J. C.; MENEGATTO, V. A.; PERON, A. P.. Strictly positive definite kernels on a product of circles. POSITIVITY, v. 21, n. 1, p. 329-342, . (12/22161-3, 14/25796-5, 14/00277-5)

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