In this thesis, we study adaptive signal decomposition in Hardy space H2 (D). It is a continuation of AFD (Adaptive Fourier Decomposition) theory, which was proposed by Prof. Tao Qian recently. The core idea of AFD is to decompose an analytic signal into basic blocks called mono-components, which have positive instantaneous frequencies. AFD originated from significant observation of Blaschke product and the “Maximal Selection Principle” by Prof. Qian et al. The thesis is arranged as follows. Chapter 1 is dedicated to the background of the general theory. In Chapter 2, we briefly recall Hardy space H2 (D) and basic signal processing techniques such as Matching Pursuit, Basis Pursuit and AFD. Based on Takenaka-Malmquist (TM) system, AFD finds approximations by consecutively selecting the poles for the TM basis functions in the sense of energy. We also introduce our original work of nonharmonic system {e iλt}λ∈C and its application to solving differential equations, which is also Matching-Pursuit-derived method. In Chapter 3, we summarize the recent theory of Compressed Sensing technique. Introduced in 2005 by David Donoho and the group of Terence Tao, Emmanuel Candes and Justin Romberg, the Compressed Sensing theory has attracted much attention in mathematics and in the engineering fields. It gives sparse recovery of signals and images in the sense of probability. However, the Null Space Property (NSP) and the Restricted Isometry Property (RIP) are difficult to test. The situation has not been improved until the random matrix theory was taken into consideration. The latest study on sparse representations via dictionaries by Compressed Sensing is discussed. In Chapter 4, we mainly analyze the asymptotic singular values distribution of reproducing kernel dictionary. These results come from our original papers in this field. They explain why the reproducing kernel dictionary is suitable for sparse reconstruction of Hardy signals. Chapter 5 is dedicated to the application of our theory. We give two examples to illustrate our algorithm. In addition, we apply our method to system identification which is also an aim of AFD.