Multi-window dilation-and-modulation frames on the half real line  

作　　者：Yunzhang Li Wei Zhang 

机构地区：[1]College of Applied Sciences,Beijing University of Technology,Beijing 100124,China [2]School of Mathematics and Information Sciences,Henan University of Economics and Law,Zhengzhou 450046,China

出　　处：《Science China Mathematics》2020年第12期2423-2438,共16页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China(Grant No.11271037)。

摘　　要：Wavelet and Gabor systems are based on translation-and-dilation and translation-and-modulation operators,respectively,and have been studied extensively.However,dilation-and-modulation systems cannot be derived from wavelet or Gabor systems.This study aims to investigate a class of dilation-and-modulation systems in the causal signal space L^2(R+).L^2(R+)can be identified as a subspace of L^2(R),which consists of all L^2(R)-functions supported on R+but not closed under the Fourier transform.Therefore,the Fourier transform method does not work in L^2(R+).Herein,we introduce the notion ofΘa-transform in L^2(R+)and characterize the dilation-and-modulation frames and dual frames in L^2(R+)using theΘa-transform;and present an explicit expression of all duals with the same structure for a general dilation-and-modulation frame for L^2(R+).Furthermore,it has been proven that an arbitrary frame of this form is always nonredundant whenever the number of the generators is 1 and is always redundant whenever the number is greater than 1.Finally,some examples are provided to illustrate the generality of our results.

关 键 词：FRAME wavelet frame Gabor frame dilation-and-modulation frame multi-window dilation-and-modulation frame 

分 类 号：O174[理学—数学]