Discrete Bernoulli Polynomials and the Best Constant of the Discrete Sobolev Inequality
Discrete Bernoulli Polynomials and the Best Constant of the Discrete Sobolev Inequality
复制标题
离散伯努利多项式和离散索博列夫不等式的最佳常数
DOI：
10.1619/fesi.51.307
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发表时间：
2008
期刊：
影响因子：
--
通讯作者：
Kohtaro Watanabe
中科院分区：
文献类型：
--
作者：
A. Nagai;Y. Kametaka;Hiroyuki Yamagishi;Kazuo Takemura;Kohtaro Watanabe
在希尔伯特空间${\\bf C}_0^N=\\biggl\\{{\\bf u}={}^t(u(0),\\cdots,u(N-1)) \\in {\\bf C}^N \\,\\biggr|\\, \\displaystyle\\sum_{i=0}^{N-1}u(i)=0\\biggr\\}中，导出了具有合适内积的Sobolev不等式的离散形式$。离散Sobolev不等式的最佳常数和最佳函数也由再现核理论得到，并用著名的伯努利多项式的离散类似物表示。讨论了离散伯努利多项式的一些有趣性质。
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.