Two-Dimensional Adaptive Fourier Decomposition

One-dimensional adaptive Fourier decomposition, abbreviated as 1-D AFD, or AFD, is an adaptive representation of a physically realizable signal into a linear combination of parameterized Szeg\"o and higher order Szeg\"o kernels of the context. In the present paper we study multi-dimensional AFDs based on multivariate complex Hardy spaces theory. We proceed with two approaches of which one uses Product-TM Systems; and the other uses Product-Szeg\"o Dictionaries. With the Product-TM Systems approach we prove that at each selection of a pair of parameters the maximal energy may be attained, and, accordingly, we prove the convergence. With the Product-Szeg\"o dictionary approach we show that Pure Greedy Algorithm is applicable. We next introduce a new type of greedy algorithm, called Pre-Orthogonal Greedy Algorithm (P-OGA). We prove its convergence and convergence rate estimation, allowing a weak type version of P-OGA as well. The convergence rate estimation of the proposed P-OGA evidences its advantage over Orthogonal Greedy Algorithm (OGA). In the last part we analyze P-OGA in depth and introduce the concept P-OGA-Induced Complete Dictionary, abbreviated as Complete Dictionary . We show that with the Complete Dictionary P-OGA is applicable to the Hardy $H^2$ space on $2$-torus.