Vol. 56, No. 1, 1975

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Growth estimates for the singular values of square-integrable kernels

James Alan Cochran

Vol. 56 (1975), No. 1, 51–58

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 μn 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∕μn)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
Received: 3 November 1973
Published: 1 January 1975
James Alan Cochran