Differentiable positive definite kernels on two-point homogeneous spaces

In this work we study continuous kernels on compact two-point homogeneous spaces which are positive definite and zonal (isotropic). Such kernels were characterized by R. Gangolli some forty years ago and are very useful for solving scattered data interpolation problems on the spaces. In the case the space is the d-dimensional unit sphere, J. Ziegel showed in 2013 that the radial part of a continuous positive definite and zonal kernel is continuously differentiable up to order ⌊(d−1)/2⌋ in the interior of its domain. The main issue here is to obtain a similar result for all the other compact two-point homogeneous spaces.CNPq (grant 141908/2015-7)FAPESP (grant 2014/00277-5

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Não disponívelThis work is composed by two parts. In the first one, we relate some basic facts about the invariance theory for non-autonomous functional differential equations of retarded type. In the last one, by using Liapunov\'s functionals and invariance properties we obtain a criterion of instability for non-autonomous functional differential equations of retarded type

Not available

Não disponívelThis work is composed by two parts. In the first one, we relate some basic facts about the invariance theory for non-autonomous functional differential equations of retarded type. In the last one, by using Liapunov\'s functionals and invariance properties we obtain a criterion of instability for non-autonomous functional differential equations of retarded type

Weighted Fourier–Laplace transforms in reproducing kernel Hilbert spaces on the sphere

We study the action of a weighted Fourier–Laplace transform on the functions in the reproducing kernel Hilbert space (RKHS) associated with a positive definite kernel on the sphere. After defining a notion of smoothness implied by the transform, we show that smoothness of the kernel implies the same smoothness for the generating elements (spherical harmonics) in the Mercer expansion of the kernel. We prove a reproducing property for the weighted Fourier–Laplace transform of the functions in the RKHS and embed the RKHS into spaces of smooth functions. Some relevant properties of the embedding are considered, including compactness and boundedness. The approach taken in the paper includes two important notions of differentiability characterized by weighted Fourier–Laplace transforms: fractional derivatives and Laplace–Beltrami derivatives.Fundação de Amparo à Pesquisa do Estado de São Paulo, processo n. 2008/57085-0 e 2010/19734-

Eigenvalue decay of integral operators generated by power series-like kernels

We deduce decay rates for eigenvalues of integral operators generated by power series-like kernels on a subset X of either 'R POT.Q' or 'C POT.Q'. A power series-like kernel is a Mercer kernel having a series expansion based on an orthogonal family '{'F IND.α'} IND.α∈'Z POT.Q IND.+'' in 'L POT.2'(X, μ), in which μ is a complete measure on X . As so, we show that the eigenvalues of the integral operators are given by an explicit formula defined by the coefficients in the series expansion of the kernel and the elements of the orthogonal family. The inequalities and, in particular, the decay rates for the sequence of eigenvalues are obtained from decay assumptions on the sequence of coefficients in the expansion of the kernel and on the sequence '{ll'f IND.α'll } IND.α∈'Z POT.Q IND.+''.FAPESP-Brasil (grants # 2010/00478-0 and # 2010/19734-6

Sharp estimates for eigenvalues of integral operators\ud generated by dot product kernels on the sphere

We obtain explicit formulas for the eigenvalues of integral operators generated by continuous dot product kernels defined on the sphere via the usual gamma function. Using them, we present both, a procedure to\ud describe sharp bounds for the eigenvalues and their asymptotic behavior near 0. We illustrate our results with examples, among them the integral operator generated by a Gaussian kernel. Finally, we sketch complex\ud versions of our results to cover the cases when the sphere sits in a Hermitian space.FAPESP-Brasil, processo n. 2010/00478-0 e 2010/19734-

Reproducing kernel Hilbert spaces associated with kernels on topological spaces

We analyze reproducing kernel Hilbert spaces of positive definite kernels on a topological space X being either first countable or locally compact. The results include versions of Mercer's theorem and theorems on the embedding of these spaces into spaces of continuous and square integrable functions.FAPEMIG [APQ-03911-10]FAPESP [2010/19734-6

Strictly positive definite kernels on the torus

We determine a necessary and sufficient condition for the strict positive definiteness of a continuous and positive definite kernel on the m-dimensional torus, enhancing a characterization of positive definiteness given by S. Bochner in 1933.FAPESP (grant 2014/00277-5