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A remark on Fefferman-Stein's inequalities.

Y. Rakotondratsimba (1998)

Collectanea Mathematica

It is proved that, for some reverse doubling weight functions, the related operator which appears in the Fefferman Stein's inequality can be taken smaller than those operators for which such an inequality is known to be true.

Integral operators generated by Mercer-like kernels on topological spaces

M. H. Castro, V. A. Menegatto, A. P. Peron (2012)

Colloquium Mathematicae

We analyze some aspects of Mercer's theory when the integral operators act on L²(X,σ), where X is a first countable topological space and σ is a non-degenerate measure. We obtain results akin to the well-known Mercer's theorem and, under a positive definiteness assumption on the generating kernel of the operator, we also deduce series representations for the kernel, traceability of the operator and an integration formula to compute the trace. In this way, we upgrade considerably similar results...

Locally convex domains of integral operators

Paweł Szeptycki (2005)

Banach Center Publications

We have shown in [1] that domains of integral operators are not in general locally convex. In the case when such a domain is locally convex we show that it is an inductive limit of L¹-spaces with weights.

Optimal domains for kernel operators on [0,∞) × [0,∞)

Olvido Delgado (2006)

Studia Mathematica

Let T be a kernel operator with values in a rearrangement invariant Banach function space X on [0,∞) and defined over simple functions on [0,∞) of bounded support. We identify the optimal domain for T (still with values in X) in terms of interpolation spaces, under appropriate conditions on the kernel and the space X. The techniques used are based on the relation between linear operators and vector measures.

Schwartz kernel theorem in algebras of generalized functions

Vincent Valmorin (2010)

Banach Center Publications

A new approach to the generalization of Schwartz’s kernel theorem to Colombeau algebras of generalized functions is given. It is based on linear maps from algebras of classical functions to algebras of generalized ones. In particular, this approach enables one to give a meaning to certain hypotheses in preceding similar work on this theorem. Results based on the properties of G -generalized functions class are given. A straightforward relationship between the classical and the generalized versions...

Spectral approach for kernel-based interpolation

Bertrand Gauthier, Xavier Bay (2012)

Annales de la faculté des sciences de Toulouse Mathématiques

We describe how the resolution of a kernel-based interpolation problem can be associated with a spectral problem. An integral operator is defined from the embedding of the considered Hilbert subspace into an auxiliary Hilbert space of square-integrable functions. We finally obtain a spectral representation of the interpolating elements which allows their approximation by spectral truncation. As an illustration, we show how this approach can be used to enforce boundary conditions in kernel-based...

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