We extend the classical Mercer theorem to reproducing kernel Hilbert spaces whose elements are functions from a measurable space X into Cn. Given a finite measure μ on X, we represent the reproducing kernel K as a 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 such that the series representing K uniformly converges to K on compact subsets of X×X, provided that the support of μ is X.

We extend the classical Mercer theorem to reproducing kernel Hilbert spaces whose elements are functions from a measurable space X into C-n. Given a finite measure mu on X, we represent the reproducing kernel K as a convergent series in terms of the eigenfunctions of a suitable compact operator depending on K and mu. 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 such that the series representing K uniformly converges to K on compact subsets of X x X, provided that the support of mu is X. (C) 2012 Elsevier Inc. All rights reserved.

An extension of Mercer theorem to matrix-valued measurable kernels

DE VITO, ERNESTO;UMANITA', VERONICA;VILLA, SILVIA
2013-01-01

Abstract

We extend the classical Mercer theorem to reproducing kernel Hilbert spaces whose elements are functions from a measurable space X into C-n. Given a finite measure mu on X, we represent the reproducing kernel K as a convergent series in terms of the eigenfunctions of a suitable compact operator depending on K and mu. 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 such that the series representing K uniformly converges to K on compact subsets of X x X, provided that the support of mu is X. (C) 2012 Elsevier Inc. All rights reserved.
2013
We extend the classical Mercer theorem to reproducing kernel Hilbert spaces whose elements are functions from a measurable space X into Cn. Given a finite measure μ on X, we represent the reproducing kernel K as a 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 such that the series representing K uniformly converges to K on compact subsets of X×X, provided that the support of μ is X.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11567/376523
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