Spatio-spectral limiting on discrete tori: adjacency invariant spaces

摘要

Discrete tori are Z_m~N thought of as vertices of graphs C_m~N whose adjacencies encode the Cartesian product structure. Space-limiting refers to truncation to a symmetric path neighborhood of the zero element and spectrum-limiting in this case refers to corresponding truncation in the isomorphic Fourier domain. Composition spatio-spectral limiting (SSL) operators are analogues of classical time and band limiting operators. Certain adjacency-invariant spaces of vectors defined on Z_m~N are shown to have bases consisting of Fourier transforms of eigenvectors of SSL operators.We show that when m = 3 or m = 4, all eigenvectors of SSL arise in this way. We study the structure of corresponding invariant spaces when m ≥ 5 and give an example to indicate that the relationship between eigenvectors of SSL and the corresponding adjacency-invariant spaces should extend to m ≥ 5.