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

Abstract

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.

Gram matrices of reproducing kernel Hilbert spaces over graphs

“…Before stating our results, we should mention the work of Nagai, Kametaka, Yamagishi, Takemura and Watanabe in [8], where some similar results have been obtained. One of differences from their research is that our interest is analysis of graphs from RKHS point of view.…”

The Best Constant of Discrete Sobolev Inequality on a Weighted Truncated Tetrahedron

“…Consider the truncated tetrahedron T4. It has 12 vertices, and let us number the vertices 0, 1, …, 11 as in Figure 1, similar to [12] , (10,11), (11,9) . e = 1 e is the set of original edges of R4, and 2 e is the set of edges of T4 created by the truncation.…”

The best constant of discrete Sobolev inequality on 1812 C60 fullerene isomers

“…The basic idea for the best constant of the discrete Sobolev inequality can be seen in, for example, [2][3][4][5][6][7][8][9][10][11]. However, for the sake of self-containedness we give a proof.…”

“…Our study of Sobolev inequality starts with [1], in which we considered the best constant of Sobolev inequality in an n-dimensional Euclidean space. Afterwards, we treated discrete Sobolev inequalities and found their best constants on polygons [2][3][4][5], polyhedra [6][7][8] and truncated polyhedra [9][10][11].…”