Inversion Formulas for the Discretized Hilbert Transform on the Unit Circle

Abstract

A discrete version of the Hilbert transform on the unit circle is considered. Its Moore--Penrose inverse with respect to suitable scalar products is derived for different side conditions. Furthermore, stability of the pseudo-inverse is studied. These results allow the efficient computation of approximate solutions of singular integral equations with Hilbert kernel. Furthermore, the stability analysis of such methods becomes much easier even for graded meshes which are useful for weakly singular solutions.