An extension of Mercer’s theory to L p

An extension of Mercer’s theory to L p Let X be a topological space, either locally compact or first countable, endowed with a strictly positive measure ν and $${\mathcal{K}:L^2(X,\nu)\to L^2(X,\nu)}$$ an integral operator generated by a Mercer like kernel K. In this paper we extend Mercer’s theory for K and $${\mathcal{K}}$$ under the assumption that the function $${x\in X\to K(x,x)}$$ belongs to some L p/2(X, ν), p ≥ 1. In particular, we obtain series representations for K and some powers of $${\mathcal{K}}$$ , with convergence in the p-mean, and show that the range of certain powers of $${\mathcal{K}}$$ contains continuous functions only. These results are used to estimate the approximation numbers of a modified version of $${\mathcal{K}}$$ acting on L p (X, ν). Positivity Springer Journals

An extension of Mercer’s theory to L p

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SP Birkhäuser Verlag Basel
Copyright © 2011 by Springer Basel AG
Mathematics; Calculus of Variations and Optimal Control; Optimization; Fourier Analysis; Operator Theory; Econometrics; Potential Theory
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