An extension of Mercer theorem to vector-valued measurable kernels

2011/10/18 by De Vito, Ernesto, Umanita`, Veronica, Villa, Silvia
#46E22 #47A70 #47B32 #47B34 #47G10 #68T05 #FOS: Mathematics #Functional Analysis (math.FA) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1110.4017

Abstract

We extend the classical Mercer theorem to reproducing kernel Hilbert spaces whose elements are functions from a measurable space Xinto \mathbb Cⁿ. Given a finite measure μ on X, we represent the reproducing kernel K as convergent series in terms of the eigenfunctions of a suitable compact operator depending on K and μ. Our result holds under the mild assumption that K is measurable and the associated Hilbert space is separable. Furthermore, we show that X has a natural second countable topology with respect to which the eigenfunctions are continuous and the series representing K uniformly converges to K on any compact subsets of X× X, provided that the support of μ is X.

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