Variational Splines and Paley--Wiener Spaces on Combinatorial Graphs

TL;DR

This paper introduces variational splines and Paley-Wiener spaces on combinatorial graphs, establishing their properties and providing a reconstruction algorithm for Paley-Wiener functions based on uniqueness sets.

Contribution

It defines and analyzes variational splines and Paley-Wiener spaces on graphs, proving existence, uniqueness, and developing a reconstruction method for these functions.

Findings

Existence and uniqueness of interpolating variational splines on graphs

Development of a reconstruction algorithm for Paley-Wiener functions

Extension of classical analysis concepts to combinatorial graphs

Abstract

Notions of interpolating variational splines and Paley-Wiener spaces are introduced on a combinatorial graph G. Both of these definitions explore existence of a combinatorial Laplace operator onG. The existence and uniqueness of interpolating variational splines on a graph is shown. As an application of variational splines, the paper presents a reconstruction algorithm of Paley-Wiener functions on graphs from their uniqueness sets.