Time-frequency analysis and nonstationary filtering

We illustrate several adaptive algorithms for the solutions of bi-infinite singular linear systems. Such algorithms are realized from exact iterative schemes (e.g., Richardson, steepest-descent methods, matching pursuit) by finite dimensional approximations of each iteration, performed with a greedy approach. We show that these algorithms are convergent and optimal with respect to certain sparseness classes of vectors as soon as the system matrix has sufficient off-diagonal decay. Bi-infinite linear systems of this type typically arise in the solution of functional operator equations (e.g., integral and differential equations) by discretization with respect to frames, i.e., stable, complete, and redundant expansions. We present applications in signal processing/transmission and PDE.