Average sampling and average splines on combinatorial graphs

Abstract

For Paley-Wiener functions on weighted combinatorial finite or infinite graphs we develop a weighted sampling theory in which samples are defined as inner products with weight functions (measuring devices). Three reconstruction methods are suggested. The first two of them are using language of dual Hilbert frames and the so-called frame algorithm respectively. The third one is using the so-called weighted variational interpolating splines which are constructed in the setting of combinatorial graphs. This development requires a new set of Poincaré-type inequalities which we prove for functions on combinatorial graphs.

Blue-Noise Sampling on Graphs

“…In particular, it will be shown in the following sections that blue-noise sampling patterns indeed promote high values of these constants. Recently Pesenson [27] introduced results that characterize the representation of a band limited signal in terms of induced subgraphs obtained from partitions of V (G) that cover V (G). This statement can be summarized in the following theorem.…”

“…Recently Pesenson [27] introduced results that characterize the representation of a band limited signal in terms of induced subgraphs obtained from partitions of V (G) that cover V (G). This statement can be summarized in the following theorem.…”

“…Theorem 3 (5.1,6.1,6.2 [27]) Let G be a connected finite or infinite and countable graph. Suppose that P = {V (Ω j )} j=|P| j=1 is a disjoint cover of V (G) by connected and finite subgraphs Ω j .…”