Countable Vector Function Bases And Characterization Of (Ω, μ)-frames

Wavelet analysis as a more and more perfect new theory is applied expansively in scientific research.Frame theory is a very important part of Wavelet analysis, and the conception of frame is brought out by Duffin and Schaefer in 1952 when they studied the nonlinear harmonic analysis.The study of frame has many practical bearings,which is applied expansively to signal processing,digital compression,sample theory etc.In frame theories;(Ω,μ)-frame is a more abstract but more expansive and directive part.This paper emphasized on the characterization of(Ω,μ)-frame and the results when frames are acted on by some operators.During these studies I got some enlightenment,and raised the conceptions such as finite vector function bases,countable vector function bases and vector function bases etc,and proved some properties of them.This paper consist of four partsThe first chapter:introduction.This part mainly illustrated the development of frame in the process of wavelet analysis research;in the meantime we introduced the direction of recent study of frame.The second part:background knowledge.This chapter mainly introduced the knowledge which would be used in the following chapter,concerning the theories of vector functions' limitation,continuation,integration etc.The third part:vector function bases.This part we introduce the conception of vector function bases.We first introduce the idea of limited vector function bases,then countable vector function bases.The fourth part:The characterization of(Ω,μ)-frame,we characterized (Ω,μ)-frame in some different walks,and studied the influences to frames after being acted on by some operators.