# A Hilbert Space Theory of Generalized Graph Signal Processing

## TL;DR

This paper introduces a comprehensive Hilbert space framework for generalized graph signal processing, extending traditional methods to infinite-dimensional spaces and integrating continuous domain analysis.

## Contribution

It develops a unified theoretical framework using functional analysis that broadens GSP to include continuous domains and infinite-dimensional Hilbert spaces.

## Key findings

- Established a Fourier-like transform for generalized GSP.
- Developed theory for filtering and sampling in the new framework.
- Unified discrete and continuous GSP approaches.

## Abstract

Graph signal processing (GSP) has become an important tool in many areas such as image processing, networking learning and analysis of social network data. In this paper, we propose a broader framework that not only encompasses traditional GSP as a special case, but also includes a hybrid framework of graph and classical signal processing over a continuous domain. Our framework relies extensively on concepts and tools from functional analysis to generalize traditional GSP to graph signals in a separable Hilbert space with infinite dimensions. We develop a concept analogous to Fourier transform for generalized GSP and the theory of filtering and sampling such signals.