Sampling Theorems In Wavelet Subspaces

In digital signal and image processing and digital communications, a continuous signal is usually represented and processed by using its discrete samples.Then how to represent a continuous signal in term of a discrete sequence is a fundamental problem. For a bandlimited signal of finite energy, it is completely characterized by its samples.by the famous classical Shannon sampling theorem. Shannon sampling theorem's presenting displayed a history milestone purpose for some theorems about image processing .Afterwards,some scholars structure miscellaneous sampling theorems based on Shannon sampling theorem ,from sampling theorems about one dimension signal to sampling theorems about many dimensions signal;from sampling theorems about definited signal to sampling theorems about generalized stable signal or non-stable signal;from sampling theorems about bandlimited signal to sampling theorems about non- bandlimited signal;from uniformly spaced sampling sequence to non-uniformly spaced sampling sequence and so on.In this paper we study sampling theorem by using frames thory.This paper consists of four chapters.In chapterl,we introduce the development of sampling theorem and the popular ways in studying sampling.In chapter2, the irregular sampling theorem on the Riesz bases is extend onto the frame;we find a necessary and sufficient condition for irregular sampling in wavelet subspace by using the frame theory.In chapter3, the uni-wavelet is constructed by multiwavelets with multiplicity r. Based on uni-wavelet generated by our method the Shannon type sampling theorem on the multiwavelets subspace is presented. Furthermore the paper provides one algorithm to estimate the error when the theorem is applied to calculate concrete signals.In chapter4,regular and irregular sampling formulas in V₀(?)L²(R²) are obtained by transforming ,via T .expansions in L² ((0,1)²) with respect to some appropriate Riesz bases.