For many water quality studies,a data analyst,or modeler, may need to know the spatio-temproal patterns of the data sets and their relationships in both pre-processing and post-processing. Geographic Information Syste...For many water quality studies,a data analyst,or modeler, may need to know the spatio-temproal patterns of the data sets and their relationships in both pre-processing and post-processing. Geographic Information System(GIS) can provide an exploratory spat展开更多
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 wa...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.展开更多
文摘For many water quality studies,a data analyst,or modeler, may need to know the spatio-temproal patterns of the data sets and their relationships in both pre-processing and post-processing. Geographic Information System(GIS) can provide an exploratory spat
基金supported by National Natural Science Foundation of China(Grant No.11271037)。
文摘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.
