The Explicit Solution of Elastodynamical Diffraction Problems by Symbol Factorization

. Since S. G. MEI.rN introduced the conept of the symbol of a singular integral operator 50 years ago [24], mathematicians working in various fields realized the importance of the fact that the structure of problems governed by convolutional 'type equations reflects in properties of the Fourier symbol function' or matrix function, respectively. Wiener-Hopf equations and systems of them 'represent one of those fields Their nature and explicit solution is directly connected with the factorization of the Fourier symbol, see the famous papers by . 1. GOHBERO and, M. G. KREIN '[8] up to the recent monographs by-S. G. MIKIILIN and S. PRöSSDORF [25] and others [13, 16, 271. The problems treated here yield symbols in a particular algebra of non-rational matrix function's, for which we present a constructive fdctorization procedure. The basic idea: s differ'completely from those which are used for rational matrix functions, see [4, 5, 7, 9]. We shall concentrate on four boundary. value problems which have been posed by .VD. KUPRADZE [15], but like to mention that the , method applies also to other boundary value arid transmission problems, see'[21, 29, 30] for admissible boundary operators and [1-3, 6, 10, 14, 17, 18, 31] for background.