Poincaré and Plancherel-Pólya inequalities in harmonic analysis on weighted combinatorial graphs. (English) Zbl 1307.05094
Authors’ abstract: We prove Poincaré and Plancherel-Polya inequalities for weighted \(\ell^p\)-spaces on weighted graphs in which the constants are explicitly expressed in terms of some geometric characteristics of a graph. We use a Poincaré-type inequality to obtain some new relations between geometric and spectral properties of the combinatorial Laplace operator. Several well-known graphs are considered to demonstrate that our results are reasonably sharp. The Plancherel-Polya inequalities allow for application of the frame algorithm as a method for reconstruction of Paley-Wiener functions on weighted graphs from a set of samples. The results are illustrated by developing Shannon-type sampling in the case of a line graph.
Reviewer: Chunhui Lai (Zhangzhou)
MSC:
05C22 | Signed and weighted graphs |
05C10 | Planar graphs; geometric and topological aspects of graph theory |
05C85 | Graph algorithms (graph-theoretic aspects) |
42C99 | Nontrigonometric harmonic analysis |
94A20 | Sampling theory in information and communication theory |
94A12 | Signal theory (characterization, reconstruction, filtering, etc.) |