RIEMANN ZETA FUNCTION, BERNOULLI POLYNOMIALS AND THE BEST CONSTANT OF SOBOLEV INEQUALITY

Abstract

Green function for periodic boundary value problem of 2M-th order ordinary differential equation is found by symmetric orthogonalization method under a suitable solvability condition. As an application, the best constants and the best functions of the Sobolev inequalities in a certain series of Hilbert spaces are found and expressed by means of the well-known Bernoulli polynomials. This result has clarified the variational meaning of the special values ζ(2M) (M = 1, 2, 3, · · · ) of Riemann zeta function ζ(z).