Numerical analysis for one-dimensional Cauchy singular integral equations

摘要

The paper presents a selection of modern results concerning the numerical analysis of one-dimensional Cauchy singular integral equations, in particular the stability of operator sequences associated with different projection methods. The aim of the paper is to show the main ideas and approaches, such as the concept of transforming the question of the stability of an operator sequence into an invertibility problem in a certain Banach algebra or the concept of certain scales of weighted Besov spaces to prove convergence rates of the sequence of the approximate solutions. Moreover, computational aspects, in particular the construction of fast algorithms, are discussed.