Frames and Bases of Translates of Signals on Undirected Graphs

Rabeetha Velsamy; Radha Ramakrishnan

paper

Frames and Bases of Translates of Signals on Undirected Graphs

arXiv:2603.21026

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.