Regular boundary value conditions for ordinary differential operators.

Abstract

Introduction. The Birkhoff theory for linear differential operators of order m treats operators determined by a nonselfadjoint differential expression and twopoint boundary conditions. In his classic of 1908 [1], Birkhoff gave determinantal inequalities, involving the leading coefficients in the boundary forms, from which asymptotic eigenfunction estimates and an expansion theorem follow. Stone [14] then showed that the Birkhoff expansion is in a sense equivalent to the Fourier expansion. In the present paper we focus on a question of Kamke [5, p. 197], and show that if m is even, selfadjoint boundary conditions meet the determinantal inequalities (which are somewhat simplified in the process). This result is not unexpected, for if