The Discussion About The Frame Wavelet Sets

Wavelet analysis is a very widely used and rapidly developed mathematical methods of analysis in recent years, plays an important role in theory and in prac-tical application, It has obtained the huge achievement in function theory, operator theory, partial differential equations, nonlinear analysis, numerical analysis and im-age processing, signal transmission, data compression, edge detection and so on. Wavelet analysis is the result of the mathematicians, physicists and engineers’s hardworking, the breakthrough development of multiple harmonic analysis. Along with the rapid development of wavelet analysis, the frame also gets more and more attention by people, the framework is an extension of the Riesz base, is proposed by Duffin and Schacffer while studying the Fourier series in1952. Since wavelet basis has the properties of moment, smoothness, attenuation and so on, and can do a local analysis, can provide unconditional base for much common functional space, the wavelet coefficient can depict function space, etc. Many scholars studied the construction method of wavelet base, constructing wavelet set especially in the frequency domain is an important method, and frame wavelet is the generalization of the wavelet in L2(Rd), has greater flexibility than the wavelet. With the deeper research of wavelet framework, many scholars have studied a special kind of wavelet (Ψ=1/√2π×E, E is a lebesgue measurable set,Ψ is the Fourier transform of Ψ), people started researching some measurable set by depicting a measurable set to describe a set of frame wavelets. The research of frame wavelet set, of course, has just started, there are still many problems to need further research.This article mainly talked about the existence of frame wavelet set, and a characterization of frame wavelet set, and some conclusions are obtained. This article mainly divides into four parts:The first chapter, is an introduction, which is briefly introduced the emergence, development of wavelet analysis and research present situation of frame wavelet set.The second chapter, has discusses the the existence of frame wavelet set in L2(R), a necessary and sufficient condition of a tight frame set、a normalized tight frame set, according to a special kind of wavelet, we use the expansion translation method of the wavelet.The third chapter, mainly discusses the existence of the frame wavelet set in L2(Rd), a necessary and sufficient condition of a tight frame set、a normalized tight frame set, also gives some conclusions.The fourth chapter, talks about the frame wavelet set of a reducing subspace in L2(Rd), a necessary and sufficient condition of a tight frame set、a normalized tight frame set of a reducing subspace, also gives some examples.