Homogeneous Approximation Problem And Dynamical Sampling Problem

Wavelet transform,linear canonical transform and special affine Fourier transform are all developed by the evolution of Fourier transform,it is important to study the homogeneous approximation property of wavelet transform,linear canonical transform and special affine Fourier transform in the application of sampling theory In this paper,the homogeneous approximation of finite interval continuous wavelet transform,the dynamic sampling of linear canonical transform and special affine Fourier transform are discussed and we get the corresponding resultIn the first chapter,the main results of this paper is introduced firstly,and then the development process of wavelet transform,linear canonical transform and affine Fourier transform are introduced.By introducing the property of homogeneous approximation and the concept of dynamic sampling,the following problems are discussed.In the second chapter,we study the HAP for the continuous wavelet transform when the integral intervals of two variables are both finite.If the pointwise HAP holds,we prove that the function to be reconstructed must be 0 when ?1=?2.This is very different from the infinite interval.For ?1??2,some necessary conditions have been given by us.In the third chapter,we first give the definitions of linear canonical transform and special affine Fourier transform,then we introduce the convolution property of them.We propose a method to recover signals in sequence space or translation invariant space by using their dynamic sampling values.Finally,in the linear canonical transform domain and the special affine Fourier transform domain,we obtain sufficient and necessary conditions for the stable reconstruction of signals in sequence space and translation invariant space.