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

MR2440944 Atsushi NAGAI, Yoshinori KAMETAKA, Hiroyuki YAMAGISHI, Kazuo TAKEMURA and Kohtaro WATANABE Atsushi NAGAI, Yoshinori KAMETAKA, Hiroyuki YAMAGISHI, Kazuo TAKEMURA and Kohtaro WATANABE Discrete Bernoulli Polynomials and the Best Constant of the Discrete Sobolev Inequality Funkcialaj Ekvacioj. Serio Internacia 51 2008 307--327 http://fe.math.kobe-u.ac.jp/FE/FullPapers/51-2/51_307.pdf http://www.ams.org/mathscinet-getitem?mr=MR2440944 A discrete version of the Sobolev inequalty in the Hilbert space<br /> ${\Bbb C}_0^N=\biggl\{\bfu={ }^t(u(0),\cdots,u(N-1))\in {\Bbb C}^N \,\biggr|\,\ \displaystyle\sum_{i=0}^{N-1}u(i)=0\biggr\},$<br /> 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. Sobolev inequality, Discrete, Bernoulli polynomials, Reproducing kernel, Penrose-Moore generalized inverse matrix. 11B68, 46E39. 51-307 2008 Discrete Bernoulli Polynomials and the Best Constant of the Discrete Sobolev Inequality Atsushi NAGAI, Yoshinori KAMETAKA, Hiroyuki YAMAGISHI, Kazuo TAKEMURA and Kohtaro WATANABE Atsushi NAGAI, Yoshinori KAMETAKA, Hiroyuki YAMAGISHI, Kazuo TAKEMURA and Kohtaro WATANABE

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