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Two types of discrete Sobolev inequalities on a weighted Toeplitz graph. (English) Zbl 1343.05097

Summary: In this paper, two types of discrete Sobolev inequalities that correspond to the generalized graph Laplacian \(A\) on a weighted Toeplitz graph are obtained. The sharp constants \(C_0(a)\) and \(C_0\) are calculated using the Green matrix \(\mathrm{G}(a) = (\mathrm{A} + a \mathrm{I})^{- 1}(0 < a < \infty)\) and pseudo-Green matrix \(\mathrm{G}_\ast =\mathrm{A}^{\operatorname{\dagger}}\) (Penrose-Moore generalized inverse matrix of \(A\)). The sharp constants are expressed as reciprocals of the harmonic mean corresponding to eigenvalues of each matrix \(\mathrm{A} + a \mathrm{I}\) and \(A\) except an eigenvalue 0.

MSC:

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
41A44 Best constants in approximation theory
46E39 Sobolev (and similar kinds of) spaces of functions of discrete variables
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