Convergence of Solutions to the Classic Dual Cosine Equation

摘要

Dual orthogonal series are a primary tool in the analysis of mixed boundary value problems for the potential in geometries conforming to separability conditions for the Laplace operator. Different formal solutions to the classic dual cosine equation, in the form of multiple singular integrals, have been developed by many authors, especially in connection with the study of brittle cracks. Recently an algorithmic representation for these solutions was developed to facilitate the writing of computer software. On the basis of this representation a rigorous proof of convergence for input functions of bounded variation is established. The proof utilizes only classical results in the theory of orthogonal expansions. In the first equation of the dual pair there is ordinary convergence, and in the second there is Abel-Poisson convergence which is not, in general, reducible to ordinary convergence.