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Convolutions, integral transforms and integral equations by means of the theory of reproducing kernels

Castro L. P., Saitoh S., Tuan N. M.

Opuscula Mathematica

2012

Vol. 32, no. 4

633-646

This paper introduces a general concept of convolutions by means of the theory of reproducing kernels which turns out to be useful for several concrete examples and applications. Consequent properties are exposed (including, in particular, associated norm inequalities).

The reactance wave diffraction problem by a strip in a scale of Bessel potential spaces

Castro L. P., Natroshvili D.

Opuscula Mathematica

2006

Vol. 26, no. 2

289-303

We consider a boundary-transmission problem for the Helmholtz equation, in a Bessel potential space setting, which arises within the context of wave diffraction theory. The boundary under consideration consists of a strip, and certain reactance conditions are assumed on it. Operator theoretical methods are used to deal with the problem and, as a consequence, several convolution type operators are constructed and associated to the problem. At the end, the well-posedness of the problem is shown for a range of regularity orders of the Bessel potential spaces, and for a set of possible reactance numbers (dependent on the wave number).