About new approach to solution of Riemann's boundary value problem with condition on the half-line in case of infinite index (Q2825960)

In this work a new approach is proposed to solve the homogeneous Riemann boundary value problem with infinite index for the half-line. The method is based on a reduction to a corresponding problem with finite index for the real axis: find a function \(\Phi(z)\) analytic and bounded on the complex plane \(z\) cut along the positive real semi-axis, \(L^+\), satisfying \(\Phi^+(t)=G(t) \Phi^-(t)\), \(t\in L^+\), where \(\Phi^+(t)\) and \(\Phi^-(t)\) are the boundary values of \(\Phi(z)\) on \(L^+\), from the left and the right, respectively, \(G(t)\) is a given function such that \(\ln|G(t)|\) is Hölder continuous and \(\mathrm{arg}\,G(t)=\nu^- t^{\rho}+\nu(t)\), \(t\in L^+\), where \(\nu^-,\rho\) are given numbers such that \(\nu^->0\), \(1/2<\rho<1 \) and \(\nu(t)\) is a Hölder continuous function.