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Frames and Bases of Translates of Signals on Undirected Graphs

Rabeetha Velsamy, Radha Ramakrishnan

Abstract

We study a shift invariant space on an undirected graphs $G$ having $N$ vertices. We obtain a characterization theorem for a system of generalized translates $\{T_{i}g : 1\leq i\leq N\}$, for $g\in C^N$, to form an orthonormal basis. Moreover, we find a necessary and sufficient condition for the system $\{T_{i}g : 1\leq i\leq m\}$, $m\leq N$, to form a linearly independent set and an orthonormal set. Further, we obtain a characterization result for a system of generalized translates which is generated by multiple generators $g_{1},...,g_{M}$ to form a frame for $C^N$. In particular, we deduce similar results for the system $\{T_{i}M_{s}g : 1\leq i,s\leq N\}$ with modulation $M_{s}$ and the spectral graph wavelet system. We also provide an illustration for the spectral graph wavelet system.

Frames and Bases of Translates of Signals on Undirected Graphs

Abstract

We study a shift invariant space on an undirected graphs having vertices. We obtain a characterization theorem for a system of generalized translates , for , to form an orthonormal basis. Moreover, we find a necessary and sufficient condition for the system , , to form a linearly independent set and an orthonormal set. Further, we obtain a characterization result for a system of generalized translates which is generated by multiple generators to form a frame for . In particular, we deduce similar results for the system with modulation and the spectral graph wavelet system. We also provide an illustration for the spectral graph wavelet system.
Paper Structure (6 sections, 15 theorems, 49 equations, 1 figure)

This paper contains 6 sections, 15 theorems, 49 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and Background
  3. Shift invariant spaces on graphs
  4. Linear independence and orthonormality for a subcollection of a system of generalized translates
  5. Shift invariant spaces with multiple generators
  6. Acknowlegdement

Key Result

Theorem 2.4

Let $\{f_{k} : k\in \mathbb{N}\}$ and $\{g_{k} : k\in \mathbb{N}\}$ be two subsets of $\mathcal{H}$. Then the following are equivalent: When one of the above equivalent conditions is satisfied, $\{f_k:k\in \mathbb{N}\}$ and $\{g_k:k\in \mathbb{N}\}$ are dual frames for $\mathcal{H}$. If $B$ denotes an upper frame bound for $\{f_k:~k\in \mathbb{N}\}$, then $B^{-1}$ is a lower frame bound for $\{g_

Figures (1)

  • Figure 1: Figure : G

Theorems & Definitions (40)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7
  • Definition 2.8
  • Remark 2.9
  • Definition 2.10
  • ...and 30 more