First and second order differentiation operators with weight functions of variable sign
Considerable difficulty arises in the study of convergence of expansions in eigen- and associated-functions (e.a.f.) by the method of contour integration in the case of exponential growth of the resolvent for large \(|\lambda |\). At present the problem has been well-studied for differential operators with irregularly decomposing boundary conditions. The function being expanded must have some properties of the type of analyticity and this is precisely what allows one to overcome the difficulties mentioned. In the present paper we show that one also observes exponential growth of the resolvent in the case of first and second order differentiation operators when weight functions of variable sign and can overcome the difficulties caused by it for a first order differentiation operator using some functional equations which the function being expanded must satisfy, and for a second order differentiation operator using its analyticity in addition. The complexity of the arguments makes it necessary to restrict oneself for now to just these operators and moreover with just one sign change. Regular boundary problems with more complicated weight functions have been studied by many authors. Different results have been obtained by \textit{M. H. Stone} [Trans. Am. Math. Soc. 26, 335-355 (1924)] for the nonregular case with \(n = 1\).
convergence of expansions in eigen- and associated-functions
scientific article; zbMATH DE number 830683