Discrete Bernoulli Polynomials and the Best Constant of the Discrete Sobolev Inequality

概要

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A discrete version of the Sobolev inequalty in the Hilbert space${\\bf C}_0^N=\\biggl\\{{\\bf u}={ }^t(u(0),\\cdots,u(N-1)) \\in {\\bf C}^N \\,\\biggr|\\, \\displaystyle\\sum_{i=0}^{N-1}u(i)=0\\biggr\\},$which 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 well-known Bernoulli polynomials. Some interesting properties of these discrete Bernoulli polynomials are also discussed.