Author(s)

Peter Junghanns & Uwe Weber

Abstract

We consider a collocation method for Cauchy singular integral equations on an in- terval based on weighted polynomials. Using Banach algebra methods, necessary and sufficient stability conditions are given in the case of continuous coefficients. 1 Introduction The subject of the present paper is the investigation of numerical methods based on weighted polynomials for the approximate solution of singular integral equations on ( — 1,1) of the type where u is the unknown function and a, 6, / are given. All functions involved are assumed to be complex-valued. We consider equation (1.1) in the Hilbert space where the inner product is defined by (%,i')<7 == I-i 2/(^)^(^k(^)^. Here cr is a Jacobi weight, that is cr(z) = ^(z) := (1 -z)"(l + z)^ satisfying the conditions -1 < a,0 < 1. These conditions guarantee the Cauchy singular integral operator 5 defined by

Keywords