This dissertation makes contribution to both the theory and applications of function approximation in complex reproducing kernel Hilbert spaces (RKHS), especially in the classical Hardy spaces. The work in the dissertation is essentially motivated by the socalled adaptive Fourier decomposition (AFD). The results can be roughly divided into two parts: theoretical and practical parts. In Chapters 3 − 5 we extend the studies of AFD to more general situations, while in Chapters 6 − 8 we study the applications of AFD and its variations. Theoretical Part: In Chapter 3, combining the ideas of the Pre-Orthogonal Greedy Algorithm and the socalled Aveiro Method, we propose the modified Aveiro Method, called Aveiro Method under Complete Dictionary, which turns out to be identical with AFD. As applications, we show that AMUCD is applicable to the Hardy space on the unit disc H2 (D) and the Paley-Wiener space W( π h ), h > 0. In Chapter 4, similar to the one dimensional case, we obtain the AFD-type approximation in H2 (TΓ1 ). As an application, we give the rational approximation of functions in L 2 (R n ). We also explore the AFD-type approximation in H2 (TΓ). In Chapter 5 we then study two other kinds of rational approximation of functions in H2 (TΓ1 ). One is in the spirit of greedy algorithm under the introduced complete dictionary. The other is to expand functions in H2 (TΓ1 ) by product-type Blaschke products. Practical Part: In Chapter 6 we compute the Hilbert transform of real-valued signals (or functions) as a by-product of applying AFD to signal processing. Compared with the existing methods, the AFD method shows its advantage and significance. In Chapter 7 we propose a modified unwinding AFD, which avoids the drawback of the numerical computation of the Hilbert transform. Consequently, we introduce the time-frequency distribution corresponding to the proposed algorithm. In Chapter 8, we study the system identifi- cation by a discrete approach, which is based on applying the greedy algorithm with the dictionary consisting of discrete rational atoms. We show the significance of the proposed method by numerical experiments.