An extension of Mercer's theory to L-p

Author(s)

Ferreira, Jose C. ^[1] ; Menegatto, Valdir A. ^[2]

Affiliation

^[1] Univ Fed Alfenas, Alfenas, MG - Brazil

^[2] ICMC USP, Sao Carlos, SP - Brazil

• Document type: Journal article
• Source: POSITIVITY; v. 16, n. 2, p. 197-212, JUN 2012.
• Web of Science Citations: 3

Abstract

Let X be a topological space, either locally compact or first countable, endowed with a strictly positive measure nu and kappa : L-2(X, nu) -> L-2(X, nu) an integral operator generated by a Mercer like kernel K. In this paper we extend Mercer's theory for K and kappa under the assumption that the function chi is an element of X -> K (chi, chi) belongs to some L-p/2(X, nu), p >= 1. In particular, we obtain series representations for K and some powers of kappa, with convergence in the p-mean, and show that the range of certain powers of kappa contains continuous functions only. These results are used to estimate the approximation numbers of a modified version of kappa acting on L-p (X, nu). (AU)