An investigation on H∞ control in relation to adaptive decomposition of signal by Wen Mi Thesis Supervisor: Professor Tao Qian Department of Mathematics University of Macau Introduction This thesis serves as an investigation on H∞ control theory in relation to some aspects of signal analysis. It is divided into 4 chapters. Chapter 1 is a brief introduction to analytic signal and the adaptive decomposition problem of current interest. It turns out that in the study we need constructive approximation to signal. We were recommended by some experts that in control theory scientists have developed a method which is called H∞ method, and they always need to constructively approximate to an aimed function by H∞ functions in the method. In Chapter 2, we first give an introduction to general linear dynamical system on the SISO case, and then give the mathematical foundation in relation to Fourier transform and Paley-Wiener Theorem. In Chapter 3, we concentrate on the problem in SISO system, giving out the description of the feedback system, the related problem and the main course of the method, including the relaxation of the problem, changing the constraints into an interpolation conditions, and the constructive sequences. Our conclusion is in Chapter 4.