On an Expansion Problem Occurring in the Theory of Diffraction

Abstract

This paper considers an eigenfunction expansion arising in certain problems in diffraction theory involving an impedance-type boundary condition. The expansion in question is usually convergent only in part of the domain of interest and involves an infinite set of orthogonal functions which do not form a complete set, in the sense that it is not in general possible to expand a given function in terms of them no matter how well-behaved the function may be. A theorem is established which asserts the validity ot" the expansion whenever it converges. Also a more general expansion formula is obtained which is valid for all sufficiently general functions and which reduces to the basic expansion whenever the latter converges.