Takemura, Kazuo; Nagai, Atsushi; Kametaka, Yoshinori Two types of discrete Sobolev inequalities on a weighted Toeplitz graph. (English) Zbl 1343.05097 Linear Algebra Appl. 507, 344-355 (2016). 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. Cited in 3 Documents 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 Keywords:Sobolev inequality; sharp constant; Toeplitz graph; graph Laplacian; Green matrix PDFBibTeX XMLCite \textit{K. Takemura} et al., Linear Algebra Appl. 507, 344--355 (2016; Zbl 1343.05097) Full Text: DOI References: [1] Aubin, T., Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geom., 11, 4, 573-598 (1976) · Zbl 0371.46011 [2] Talenti, G., Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4), 110, 353-372 (1976) · Zbl 0353.46018 [3] Kametaka, Y.; Watanabe, K.; Nagai, A., The best constant of Sobolev inequality in an \(n\) dimensional Euclidean space, Proc. Japan Acad. Ser. A Math. Sci., 81, 3, 57-60 (2005) · Zbl 1100.46021 [4] Takemura, K.; Yamagishi, H.; Kametaka, Y.; Watanabe, K.; Nagai, A., The best constant of Sobolev inequality corresponding to a bending problem of a beam on an interval, Tsukuba J. Math., 33, 2, 253-280 (2009) · Zbl 1209.34016 [5] Takemura, K.; Kametaka, Y.; Watanabe, K.; Nagai, A.; Yamagishi, H., Sobolev type inequalities of time-periodic boundary value problems for Heaviside and Thomson cables, Bound. Value Probl., 2012, Article 95 pp. (2012) · Zbl 1282.46035 [6] Nagai, A.; Kametaka, Y.; Yamagishi, H.; Takemura, K.; Watanabe, K., Discrete Bernoulli polynomials and the best constant of the discrete Sobolev inequality, Funkcial. Ekvac., 51, 2, 307-327 (2008) · Zbl 1158.46026 [7] Yamagishi, H.; Kametaka, Y.; Watanabe, K.; Nagai, A.; Takemura, K., The best constant of three kinds of discrete Sobolev inequalities on regular polyhedron, Tokyo J. Math., 36, 1, 253-268 (2013) · Zbl 1288.46028 [8] Seto, M.; Suda, S.; Taniguchi, T., Gram matrices of reproducing kernel Hilbert spaces over graphs, Linear Algebra Appl., 445, 56-68 (2014) · Zbl 1308.46035 [9] Olshevsky, V.; Strang, G.; Zhlobich, P., Green’s matrices, Linear Algebra Appl., 432, 218-241 (2010) · Zbl 1195.15012 [10] Dal van, R.; Tijssen, G.; Tuza, Z.; van der Veen, J. A.A.; Zamfirescu, C.; Zamfirescu, T., Hamiltonian properties of Toeplitz graphs, Discrete Math., 159, 1-3, 69-81 (1996) · Zbl 0864.05060 [11] Gray, R. M., Toeplitz and Circulant Matrices: A Review (2006), Now Publishers · Zbl 1115.15021 [12] Horn, R. A.; Johnson, C. R., Matrix Analysis (2013), Cambridge University Press · Zbl 1267.15001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.