Abstract
Green function for periodic boundary value problem of 2M-th order or- dinary 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).
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