Frames and Bases of Translates of Signals on Undirected Graphs

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 $\{T_{i}g:1\le i\le N\}$, for $g\in C^N$, to form an orthonormal basis. Moreover, we find a necessary and sufficient condition for the system $\{T_{i}g:1\le i\le m\}$, $m\le N$, 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 $g_1,\mathrm{.}\mathrm{.}\mathrm{.},g_M$ to form a frame for $C^N$. In particular, we deduce similar results for the system $\{T_i{M}_{s}g:1\le i,s\le N\}$ with modulation $M_s$ and the spectral graph wavelet system. We also provide an illustration for the spectral graph wavelet system.