# Weighted sampling and weighted interpolation on combinatorial graphs

## TL;DR

This paper develops a weighted sampling theory for Paley-Wiener functions on combinatorial graphs, introducing three reconstruction methods and new Poincaré-type inequalities for these functions.

## Contribution

It introduces a novel weighted sampling framework and three reconstruction algorithms for functions on combinatorial graphs, supported by new inequalities.

## Key findings

- Three reconstruction methods using frames, algorithms, and splines.
- Development of Poincaré-type inequalities for graph functions.
- Extension of sampling theory to weighted 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 Poincar\'e-type inequalities which we prove for functions on combinatorial graphs.