In this thesis, I mainly talk about an approach to adaptive decomposition of nonlinear and nonstable signals in signal analysis. I obtain some results on analytic signals in relation to some established theories under the frame work of T.Qian; Q.H Chen and L.Q.Li(see[3].[4]). This thesis contains proofs of the Plemelj Theorem and the Bedrosian Theorem. I discuss boundary values of Hardy H² functions and prove that for a complex-valued L² function, its imaginary parts is the Hillbert transform of its real part and only if the L² function is the boundary value of a H² function. The method which I use is mainly the Fourier multiplier method. The counterpart theory in the unit disc is also studied. The outline of the thesis is as follows: Chapter 1 contains an introduction to the background knowledge of analytic signals and a survey on the Nevalina classes in the two contexts, the unit disc D and the upper-half plane C⁺. The two important theorems-the Plemelj Theorem and the Bedrosian Theorem are proved in this chapter. We include a concise introduction to Nevanlina class in relation to the Hardy space in §1.3