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
1
Kametaka, Y.; Watanabe, K.; Nagai, A.
The best constant of Sobolev inequality in an $n$ dimensional Euclidean space
Proc. Japan Acad. Ser. A
81
2005
57--60
MR2128933
2
Kametaka, Y.; Yamagishi, H.; Watanabe, K.; Nagai, A.; Takemura, K.
Riemann zeta function, Bernoulli polynomials and the best constant of Sobolev inequality
Sci. Math. Jpn.
65
2007
333--359
MR2328214
3
Komatsu, Y.
Special functions
Asakura
[in Japanese]
4
Graham, R. L.; Knuth, D. E.; Patashnik, O.
Concrete Mathematics
Addison Wesley
1994
MR1397498
5
Talenti, G.
Best constant in Sobolev inequality
Ann. Mat. Pura. Appl.
110
1976
353--372
MR0463908
6
Watanabe, K.; Kametaka, Y.; Nagai, A.; Takemura, K.; Yamagishi, H., The best constants of Sobolev and Kolomogorov type inequalities on a half line, submitted