Characterizations Of Scaling Function With Composite Dilation Multiresolution Analysis

A functionψ(x)∈L²(R)is an orthonormal wavelet provided the system{ψ_j,k(x)= 2^j/2ψ(2^jx-k)|j,k∈Z}is an orthonormal basis for L²(R).In the higher dimension space, many ways have been used to define wavelets.As well known,tensor-product wavelets are very popular.But,wavelets got in this way have finite directions,thus led to many limitations in applications.The disadvantages of wavelet analysis have made scientists try to look for better tools from various angles.Therefore,scientists engage in wavelet analysis have carried out a new revolution——Multiscale Geometry Analysis(MGA). MGA originates from wavelets but is superior to wavelets.Basic functions in MGA have good properties such as anisotropy,directionality and so on,which can solve the problems of wavelets in varying degrees.In 2004,K.Guo etal put forward the notion of wavelets with composite dilations and described the properties of wavelets with composite dilations in detail.They found that wavelets with composite dilations have many nice properties in geometry so that they have great advantages in the aspect of application.In the same way as constructing classic wavelets by multiresolution analysis,wavelets with composite dilations are also constructed by multiresolution with composite dilations. The main purpose of this paper is to discuss the properties in multiresolution analysis with composite dilations and give the characterizations of scaling function with composite dilation multiresolution analysis.The thesis is divided into five chapters.Chapter 1 briefly introduces the background of Fourier analysis,wavelet analysis and wavelet analysis with composite dilations.Chapter 2 gives the characterization of scaling function with composite dilation multiresolution analysis.The given results generalize the classic characterization of generator for multiresolution analysis.In order to prove main results,some auxiliary results are listed in Chapter 3.Chapter 4 presents the proofs of main results.In the last chapter,some examples are given.