In this paper, we present a method for constructing multivariate tight framelet packets associated with an arbitrary dilation matrix using unitary extension principles. We also prove how to construct various tight fra...In this paper, we present a method for constructing multivariate tight framelet packets associated with an arbitrary dilation matrix using unitary extension principles. We also prove how to construct various tight frames for L2 （JRa） by replacing some mother framelets.展开更多
An important tool for the construction of periodic wavelet frame with the help of extension principles was presented in the Fourier domain by Zhang and Saito [Appl. Comput. Harmon. Anal., 2008, 125： 68-186]. We exten...An important tool for the construction of periodic wavelet frame with the help of extension principles was presented in the Fourier domain by Zhang and Saito [Appl. Comput. Harmon. Anal., 2008, 125： 68-186]. We extend their results to the dilation matrix cases in two aspects. We first show that the periodization of any wavelet frame constructed by the unitary extension principle formulated by Ron and Shen is still a periodic wavelet frame under weaker conditions than that given by Zhang and Saito, and then prove that the periodization of those generated by the mixed extension principle is also a periodic wavelet frame if the scaling functions have compact supports.展开更多
A sufficient condition for affine frame with an arbitrary matrix dilation is presented. It is based on the univariate case by Shi, and generalizes the univariate results of Shi, Casazza and Christensen from one dimens...A sufficient condition for affine frame with an arbitrary matrix dilation is presented. It is based on the univariate case by Shi, and generalizes the univariate results of Shi, Casazza and Christensen from one dimension with an arbitrary real number a （a 〉 1） dilation to higher dimension with an arbitrary expansive matrix dilation.展开更多
The homogeneous approximation property （HAP） states that the number of building blocks involved in a reconstruction of a function up to some error is essentially invariant under time-scale shifts. In this paper, we ...The homogeneous approximation property （HAP） states that the number of building blocks involved in a reconstruction of a function up to some error is essentially invariant under time-scale shifts. In this paper, we show that every wavelet frame with nice wavelet function and arbitrary expansive dilation matrix possesses the HAP. Our results improve some known ones.展开更多
The refinability and approximation order of finite element multi-scale vector are discussed in [1]. But the coefficients in the conditions of approximation order of finite element multi-scale vector are incorrect ther...The refinability and approximation order of finite element multi-scale vector are discussed in [1]. But the coefficients in the conditions of approximation order of finite element multi-scale vector are incorrect there. The main purpose of this note is to make a correction of the error in the main result of [1]. These Cuefficients are very important for the properties of wavelets, such as vanishing moments and regularity.展开更多
文摘In this paper, we present a method for constructing multivariate tight framelet packets associated with an arbitrary dilation matrix using unitary extension principles. We also prove how to construct various tight frames for L2 （JRa） by replacing some mother framelets.
基金Acknowledgements The authors express their gratitude to the anonymous referees for their kind suggestions and useful comments on the original manuscript, which resulted in this final version. This work was supported by the National Natural Science Foundation of China （No. 61071189）, the Natural Science Foundation for the Education Department of Henan Province of China （No. 13A110072）, and the Natural Science Foundation of Henan University （No. 2011YBZR001）.
文摘An important tool for the construction of periodic wavelet frame with the help of extension principles was presented in the Fourier domain by Zhang and Saito [Appl. Comput. Harmon. Anal., 2008, 125： 68-186]. We extend their results to the dilation matrix cases in two aspects. We first show that the periodization of any wavelet frame constructed by the unitary extension principle formulated by Ron and Shen is still a periodic wavelet frame under weaker conditions than that given by Zhang and Saito, and then prove that the periodization of those generated by the mixed extension principle is also a periodic wavelet frame if the scaling functions have compact supports.
基金Supported by the National Natural Science Foundation of China (No.10671062)Innovation Scientists and Technicians Troop Construction Projects of Henan Province of China (No.084100510012)the Natural Science Foundation for the Education Department of Henan Province of China (No.2008B510001)
文摘A sufficient condition for affine frame with an arbitrary matrix dilation is presented. It is based on the univariate case by Shi, and generalizes the univariate results of Shi, Casazza and Christensen from one dimension with an arbitrary real number a （a 〉 1） dilation to higher dimension with an arbitrary expansive matrix dilation.
基金Supported partially by National Natural Science Foundation of China(Grant Nos.10971105and10990012)Natural Science Foundation of Tianjin(Grant No.09JCYBJC01000)
文摘The homogeneous approximation property （HAP） states that the number of building blocks involved in a reconstruction of a function up to some error is essentially invariant under time-scale shifts. In this paper, we show that every wavelet frame with nice wavelet function and arbitrary expansive dilation matrix possesses the HAP. Our results improve some known ones.
文摘The refinability and approximation order of finite element multi-scale vector are discussed in [1]. But the coefficients in the conditions of approximation order of finite element multi-scale vector are incorrect there. The main purpose of this note is to make a correction of the error in the main result of [1]. These Cuefficients are very important for the properties of wavelets, such as vanishing moments and regularity.
