Growth estimates for the singular values of square-integrable kernels

James Alan Cochran

Vol. 56 (1975), No. 1, 51–58

DOI: 10.2140/pjm.1975.56.51

Abstract

This paper is concerned with the unique correspondences which exist between the values of convergence exponents for the classical Fourier coefficients of one-variable functions satisfying various smoothness assumptions, on the one hand, and growth estimates for the singular values μ_n associated with square-integrable two-variable kernels K(x,y),a ≦ x,y ≦ b, having comparable smoothness, on the other. Extending earlier work of the author and others, precise values are given for the infimum of γ for which Σ(1∕μ_n)^γ converges when K satisfies Lipschitz conditions, Integrated Lipschitz conditions, is of bound variation, or a combination of these.

Mathematical Subject Classification 2000

Primary: 45A05

Secondary: 42A16

Milestones

Received: 3 November 1973

Published: 1 January 1975