V. A. Menegatto

A necessary and sufficient condition for strictly positive definite functions on spheres

We give a necessary and sufficient condition for the strict positive-definiteness of real and continuous functions on spheres of dimension greater than one.

Eigenvalue decay of positive integral operators on the sphere

Abstract. We obtain decay rates for singular values and eigenvalues of integral operators generated by square integrable kernels on the unit sphere in Rm+1, m ≥ 2, under assumptions on both, certain derivatives of the kernel and the integral operators generated by such derivatives. This type of problem is common in the literature but the assumptions are usually defined using standard differentiation in Rm+1. In this paper, the assumptions are all defined via the Laplace-Beltrami derivative, a concept first investigated by Rudin in the early fifties and genuinely spherical in nature. The rates we present depend on both, the differentiability order used to define the smoothness conditions and the dimension m. They are shown to be optimal.

Strictly positive definite kernels on the hilbert sphere

We completely characterize the strictly positive definite and the strictly conditionally negative definite radial continuous kernels on the real Hilbert sphere. Any functions generating such kernels, can be used in radial basis interpolation of arbitrary data on a set of points in any finite–dimensional sphere.

Integral operators on the sphere generated by positive definite smooth kernels

We consider integral operators on the unit sphere generated by positive definite kernels. Under smoothness conditions of Lipschitz-type on the kernel, we obtain a decay rate for the eigenvalues of the integral operator. The approach we have chosen is a multi-dimensional version, adapted to the spherical setting, of a known procedure used in the analysis of a similar problem for integral operators on the interval [0, 1]. In addition to spectral theory, the critical arguments in the paper involve the use of special covers of the sphere generated by quadrature formulas. The estimates themselves are comparable to others in the literature.

Strictly positive definite kernels on subsets of the complex plane

Let (z, w) @? @? x @? (zw) be a positive definite kernel and B a subset of @?. In this paper, we seek conditions in order that the restriction (z, w) @? B x B(zw) be strictly positive definite. Since this problem has been solved recently in the cases in which B is either @? or the unit circle in @?, our purpose here is twofold: to present some results we obtained when attempting to solve the problem for the above and other choices of B and to acquaint the audience with some other questions that remain. For two different classes of subsets, we completely characterize the strict positive definiteness of the kernel. We include a complete discussion of the case in which B is the unit circle of @?, making a comparison with the classical problem of strict positive definiteness on the real circle.

An extension of a theorem of Schoenberg to products of spheres

We present a characterization for the continuous, isotropic and positive definite kernels on a product of spheres along the lines of a classical result of I. J. Schoenberg on positive definiteness on a single sphere. We also discuss a few issues regarding the characterization, including topics for future investigation.

OLD AND NEW ON THE LAPLACE-BELTRAMI DERIVATIVE

The (strong) Laplace-Beltrami derivative and related operators appear quite frequently in many problems involving Fourier series, spherical harmonics, approximation of functions and smoothness. This article intends to provide an overview with updates on such operators. While most of the results concerning the Laplace-Beltrami derivative found in the literature are stated in the three-dimensional setting, our approach includes higher-dimensional spheres. We present proofs for results which are labeled as known but are just mentioned elsewhere. Spherical moduli of smoothness and decay rates for eigenvalues of integral operators are also among the covered topics.

Approximation by spherical convolution

Uniform and L[p] approximations of a given function defined on a sphere by spherical convolutions of it with suitable kernels are studied. We fill in some gaps in the theory of approximation by spherical convolution and we derive approximation results using smooth and localized kernels. We introduce a method of approximation that is, in a certain sense, comparable to the so-called moving averages method to approximate identities. Several important examples are discussed

REPRODUCING PROPERTIES OF DIFFERENTIABLE MERCER-LIKE KERNELS ON THE SPHERE

We study differentiability of functions in the reproducing kernel Hilbert space (RKHS) associated with a smooth Mercer-like kernel on the sphere. We show that differentiability up to a certain order of the kernel yields both, differentiability up to the same order of the elements in the series representation of the kernel and a series representation for the corresponding derivatives of the kernel. These facts are used to embed the RKHS into spaces of differentiable functions and to deduce reproducing properties for the derivatives of functions in the RKHS. We discuss compactness and boundedness of the embedding and some applications to Gaussian-like kernels.

Differentiable positive definite kernels on spheres

Abstract We analyze term-by-term differentiability of uniformly convergent series of the form , where Sm –1 is the unit sphere in , and {Yk } is a sequence of spherical harmonics or even more general functions. Since this class of kernels includes the continuous positive definite kernels on Sm –1, the results in this paper will show that, under certain conditions, the action of convenient differential operators on positive definite (strictly positive definite) kernels on Sm –1 generate positive definite kernels.