Mono-components From Blaschke Products And Wave Equation

The essence of Fourier analysis is to present signals by using time-frequency atoms with constant frequency.As derivatives of Fourier analysis,spectrogram,almost all time-frequency tools such as Wigner distribution,wavelet analysis fall into this framework.Un-fortunately,most of real signals are transient signals,which have time-varying frequency,namely,instantaneous frequency(IF).It is natural to approximate them by time-frequency atoms with nonlinear phase.Recently,several groups are interested in the time-frequency analysis of so called nonlinear Fourier atoms.Many theoretical results enrich the research of non-linearization of Fourier analysis.The so called nonlinear Fourier atoms are the bound-ary values of the Blaschke products on the unit disk or upper half plane.This kind of atoms have nonlinear phase(equivalently time-varying frequency),satisfies the Bedrosian identity and could be regarded as the simplest mono-components.The above research is an impor-tant advance in the nonlinearity of Fourier analysisThis thesis focuses on investigation of nonlinear Fourier atoms from point view of partial differential equations.Starting from Sturm-Liouville operators,the essential rela-tionships between wave equation and mono-components are established.The contents are as follows:firstly,the roles of some time-frequency tools from Fourier analysis in digital signal processing are recalled.From the finite Blaschke products in Hardy space on the unit disc,a class of functions ei?n(·)with nonlinear phase are introduced.They satisfy the Bedrosian identity(a product property of Hilbert transform).The Sturm-Liouville operators are naturally introduced from nonlinear Fourier atoms.Secondly,noting that the Gram-Schmidt(G-S)procedure maps the Blaschke product system {Bn:n?N} to the Takenaka-Malmquist(TM)system[1-4],different models are constructed adapting to the choice of polar coordinates.For the general system{ei?n(·):n?Z},for negative n,ei?n,(·)is defined by ei?n(·)=ei?-n(·),and will investigate a type of wave equation,which generates this system.Finally,using Sturm-Liouville operators,we study the relationship between the solution of wave equations with definite solution conditions and mono-components.