The sign-regularity properties of Green's functions for a special class of mixed boundary-value problems

Abstract

Karlin and Pinkus [5] showed that the Green’s functions for a class of mixed boundary-value problems are sign consistent of all even or all odd orders. \Vhen Karlin and Pinkus apply their results to periodic boundary conditions they recover the results of Karlin and Lee [4] in slightly weakened form. Specifically, the sign-consistency properties of the Green’s function developed in [4] hold on the square [0, 1) x [0, 1) which, in view of the periodicity of the Green’s function, is the largest domain for which such properties can hold. However, when the results in [5] are specialized to the periodic case, the sign consistency assertions of [4] are only recovered for the open square (0, I) x (0, 1). Since the analysis in [5] is particularly natural for analyzing sign consistent! of Green’s functions, it is useful to disclose how the methods in [S] can be used to obtain the results in [4] in their full strength. Essentially this requires establishing an extension of the basic total positivity properties of the kernel K(x, w) as elaborated in [2, Theorems 1 and 1’1; see Theorem 2.1 below. This extension is made in Section 2 and used in Section 4 to establish analogs of the results obtained in [4] for a class of boundary conditions which includes periodic and antiperiodic boundary conditions as well as some special boundary conditions which are “partly mixed.” Section 5 presents applications of the results of Section 4 to spline interpolation. Finally, in Section 6 the results of Karon [3] and Karlin [2] for separated boundary conditions are reformulated in the light of the results in Section 2. The notation of [I] will be used: Let A be an m x n matrix and Y < m, 12. Then