We consider integral operators defined by positive definite kernels K : X × X → C, where X is a metric space endowed with a strictly-positive measure. We update upon connections between two concepts of positive definiteness and upgrade on results related to Mercer like kernels. Under smoothness assumptions on K, we present decay rates for the eigenvalues of the integral operator, employing adapted to our purposes multidimensional versions of known techniques used to analyze similar problems in the case where X is an interval. The results cover the case when X is a subset of R m endowed with the induced Lebesgue measure and the case when X is a subset of the sphere S m endowed with the induced surface Lebesgue measure. (2000). Primary 45P05; Secondary 42A82, 45C05, 43A35, 41A99. Mathematics Subject Classification

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

An ]ntemational Joumal Available online at www.sciencedirect.com computers & ,,c,=,,c= C~o,,,=cT-mathematics with applicationsAbstract--Let (z, w) c C x C ~ f(zff;) be a positive definite kernel and B a subset of C. In this paper, we seek conditions in order that the restriction (z, w) G B x 13 ~ f(z~5) be strictly positive definite. Since this problem has been solved recently in the cases in which B is either C or the unit circle in C, 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 ease in which B is the unit circle of C, making a comparison with the classical problem of strict positive definiteness on the real circle.

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.Mathematics Subject Classifications (2010): 43A35, 33C50, 33C55, 42A10, 42A82