Über das Reduktionsverfahren für diskrete Wiener-Hopf-Gleichungen mit unstetigem Symbol

Abstract

The results of the paper complete and improve some hitherto existing results about the reduction method for discrete Wiener-Hopf equations with discontinuous symbol and, on the other hand, illustrate some new aspects of this set of problems. Thus, it is proved that if the reduction method in $l^p(1 \\leq p < \\infty)$ is applicable to the Toeplitz operators $T(a_r), a_r \\in L^{\\infty} (r = 1, \\dots, R)$ and if $a_1, \\dots, a_R$ have no common singularities (discontinuities), then it is applicable to $T(a_1 \\dots a_R)$. Furthermore, there is proved that for a large class of piecewise continuous symbols the reduction method in $L^p$ is applicable to $T(a)$ if and only if both $T(a)$ and $T(\\bar {a})$ are invertible.