Exact solutions for a class of matrix Riemann-Hilbert problems

Abstract

In contrast to scalar Riemann–Hilbert problems, a general matrix Riemann–Hilbert problem cannot be solved in terms of Sokhotskyi–Plemelj integrals. As far as the authors know, the exact solutions are known for a class of homogeneous matrix Riemann–Hilbert problems with commutative and factorable kernels. This article considered matrix Riemann–Hilbert problems in which all the partial indices are zero and the logarithms of the components of the kernels and their non-homogeneous vectors are exponential-type (equivalently, band-limited) functions. Then, it develops exact solutions for such matrix Riemann–Hilbert problems. Applications in a class of spectral factorizations and a class of the Wiener–Hopf system of integrations are given.