Frame Multiresolution Analyses And Wavelet In High Dimension Spaces

Wavelet analyses is one of the disciplines of applied mathematics whichdeveloped rapidly in recent years, Research began in the early 80s of the 20thcentury. Ithas a wide range of applications such as signal processing, imageprocessing, digital watermarking, partial differential equations, numericalcalculation of electromagnetic fields and so on, though the wavelet analysesfrom produces to present merely several dozens years.Two topics of the dissertation research about the wavelet analyses include:First, we studied the stability and the perturbation of the near-Riesz base fromframe; Second, we studied the properties of frame multiresolution analyses andwavelet in high-dimension spaces. In chapter 1, first introduced the developmenthistory and the prospects for development of the wavelet analyzes, then introducethe mainly conclusion of this article. In chapter 2, we first introduced theframe definition and properties, and then describes the properties of the Rieszbase and the near-Riesz base and their respective equivalence relations betweenthemseleves, finally we studied the stability and perturbation of the near-Rieszbase on emphasis, then obtained a new necessary and sufficient condition of theperturbation of the near-Riesz base. In chapter 3, first introduced thedefinition of frame multiresolution analyses in high-dimension space, thenobtained some properties and a necessary and sufficient condition of a set isa spectrum of frame multiresolution analyses. In chapter 4, we first introducedthe definition and properties of frame wavelet in high-dimension spaces, thenintroduced the definition of wavelet set, finally obtained a necessary andsufficient condition for a set is a wavelet set in a high-dimension spaces. Inchapter 5, we obtained a important conclusion though integer classification intwo-dimension space, use it, we obtaioned a necessary and sufficient conditionof multiresolutions analyses has the single Parseval frame wavelet in two- dimension space, and we give the express of Parseval frame wavelet.