Ana Paula Peron

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.

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.

Strictly positive definite kernels on a product of circles

We supply a Fourier characterization for the real, continuous, isotropic and strictly positive definite kernels on a product of circles. In other words, if

STRICT POSITIVE DEFINITENESS ON SPHERES VIA DISK POLYNOMIALS

S^1

Cytotoxic and genotoxic potential of liquid synthetic food flavorings evaluated alone and in combination.

S1 is the unit circle in