  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.
