Invertibility of convolution operators in problems of wave diffraction by a strip with reactance and dirichlet conditions

Abstract

This paper deals with the invertibility of convolution type operators that come from a wave diffraction problem with reactance conditions on a strip. The diffraction problem is reformulated as a single convolution type operator on a finite interval. To develop an operator constructive approach, several matrix operator identities are established between this convolution type operator and certain new Wiener-Hopf operators, and certain equivalent properties are obtained between all the related operators. Factorizations are presented for particular semi-almost periodic matrix functions and the corresponding Wiener-Hopf operators. As a result, conditions are obtained to ensure the invertibility of all the convolution type operators associated with the problem. This leads to the well-posedness of the problem including the continuous dependence on the data. In obtaining our results a major role is played by the invertibility of the convolution type operator associated with the wave diffraction by a strip with equal Dirichlet conditions on both sides of the strip, which is obtained through an analytical representation. Both problems and the corresponding operators are considered in the framework of Bessel potential spaces. © Heldermann Verlag Berlin.