Green's functions to 2-point simple type self-adjoint boundary value problems for bending of a beam under relatively strong tension on an elastic foundation are studied. We have 9 different Green's functions. All are positive-valued and have a suitable hierarchical structure.

Abstract. A discrete version of the Sobolev inequalty in the Hilbert spacewhich is equipped with a suitable inner product, is derived. The best constant and best function of the discrete Sobolev inequality are also obtained from the theory of reproducing kernels, and are expressed by means of discrete analogues of the wellknown Bernoulli polynomials. Some interesting properties of these discrete Bernoulli polynomials are also discussed.

The best constants C m,j of Sobolev embedding of H m (0, a) into C j [0, a] (0 j m − 1) are obtained. Especially, when a = ∞, these constants can be represented in a closed form.

Green function of 2-point simple-type self-adjoint boundary value problem for 4-th order linear ordinary di¤erential equation, which represents bending of a beam with the boundary condition as clamped, Dirichlet, Neumann and free. The construction of Green function needs the symmetric orthogonalization method in some cases. Green function is the reproducing kernel for suitable set of Hilbert space and inner product. As an application, the best constants of the corresponding Sobolev inequalities are expressed as the maximum of the diagonal values of Green function.