Characterization of matrix Fourier multiwavelet frames multipliers with integer dilation factor

Volume 10, Issue 9, pp 4915--4929
Publication Date: September 22, 2017 Submission Date: July 10, 2017
• 945 Views

Authors

Fengjuan Zhu - School of Mathematics and Information Science, North Minzu University, Yinchuan, 750021, China. Yongdong Huang - School of Mathematics and Information Science, North Minzu University, Yinchuan, 750021, China. Shengnan Shi - School of Mathematics and Information Science, North Minzu University, Yinchuan, 750021, China. Xiao Tan - School of Mathematics and Information Science, North Minzu University, Yinchuan, 750021, China. Juan Zhao - School of Mathematics and Information Science, North Minzu University, Yinchuan, 750021, China.

Abstract

This paper investigates matrix Fourier multiwavelet frames multipliers with dilation factor $a$. First, the definition of matrix Fourier multiwavelet frame multiplier was proposed, which is $N_1\times N$ matrices with $L^\infty$ function entries, and maps Parseval multiwavelet frames of length $N$ to Parseval multiwavelet frames of length $N_1$. Then, two sufficient conditions of matrix Fourier multiwavelet frame multiplier were given, and two necessary conditions of matrix Fourier multiwavelet frame multiplier were characterized by means of frame wavelet sets. Finally, several numerical examples were constructed. As Fourier wavelet frames multiplier, matrix Fourier multipliers can be used to derive new Parseval multiwavelet frames and can help us better understand the basic of frame theory.

Share and Cite

ISRP Style

Fengjuan Zhu, Yongdong Huang, Shengnan Shi, Xiao Tan, Juan Zhao, Characterization of matrix Fourier multiwavelet frames multipliers with integer dilation factor, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 9, 4915--4929

AMA Style

Zhu Fengjuan, Huang Yongdong, Shi Shengnan, Tan Xiao, Zhao Juan, Characterization of matrix Fourier multiwavelet frames multipliers with integer dilation factor. J. Nonlinear Sci. Appl. (2017); 10(9):4915--4929

Chicago/Turabian Style

Zhu, Fengjuan, Huang, Yongdong, Shi, Shengnan, Tan, Xiao, Zhao, Juan. "Characterization of matrix Fourier multiwavelet frames multipliers with integer dilation factor." Journal of Nonlinear Sciences and Applications, 10, no. 9 (2017): 4915--4929

Keywords

• Parseval frames multiwavelet
• Fourier multipliers
• matrix Fourier multiwavelet frames multipliers
• frame wavelet set

•  42C15
•  42C40

References

• [1] D. Bakić, I. Krishtal, E. N. Wilson, Parseval frame wavelets with $E^ (2)_ n$ -dilations, Appl. Comput. Harmon. Anal., 19 (2005), 386–431.

• [2] M. Bownik, A characterization of affine dual frames in $L^2(R^n)$, Appl. Comput. Harmon. Anal., 8 (2000), 203–221.

• [3] M. Bownik , Connectivity and density in the set of framelets, Math. Res. Lett., 14 (2007), 285–293.

• [4] M. Bownik, The closure of the set of tight frame wavelets, Acta Appl. Math., 107 (2009), 195–201.

• [5] J.-F. Cheng, D.-F. Li , The characterization of MRA E-tight frame wavelet, Acta Math. Sin. Chin. Ser., 51 (2008), 877–888.

• [6] X.-D. Dai, D. R. Larson, Wandering vectors for unitary systems and orthogonal wavelets, Mem. Amer. Math. Soc., 134 (1998), 68 pages.

• [7] X.-D. Dai, D. R. Larson, D. M. Speegle, Wavelet sets in $R^n$, J. Fourier Anal. Appl., 3 (1997), 451–456.

• [8] D.-G. Han, D. R. Larson, Wandering vector multipliers for unitary groups, Trans. Amer. Math. Soc., 352 (2001), 3347– 3370.

• [9] D.-G. Han, D. R. Larson, On the orthogonality of frames and the density and connectivity of wavelet frames, Acta Appl. Math., 107 (2009), 211–222.

• [10] D.-G. Han, D. R. Larson, Unitary systems and Bessel generator multipliers, Wavelets and multiscale analysis , Appl. Numer. Harmon. Anal., Birkhäuser /Springer, New York, (2011), 131–150.

• [11] Y.-D. Huang, N. Sun, Characterizations of A-Parseval frame wavelet, Acta Math. Sin. Chin. Ser., 54 (2011), 767–790.

• [12] Y.-D. Huang, F.-J. Zhu, Characterization of matrix Fourier multipliers for A-dilation Parseval multi-wavelet frames, Int. J. Wavelets Multiresolut. Inf. Process., 13 (2015), 22 pages.

• [13] Y.-Z. Li , On a class of bidimensional nonseparable wavelet multipliers, J. Math. Anal. Appl., 270 (2002), 543–560.

• [14] Z.-Y. Li, D.-G. Han, Matrix Fourier multipliers for Parseval multi-wavelet frames, Appl. Comput. Harmon. Anal., 35 (2013), 407–418.

• [15] Z.-Y. Li, X.-L. Shi , Parseval frame wavelet multipliers in $L^2(R^d)$, Chin. Ann. Math. Ser. B, 33 (2012), 949–960.

• [16] R. Liang , Some properties of wavelets, Thesis (Ph.D.)–The University of North Carolina at Charlotte, ProQuest LLC, Ann Arbor, MI (1998)

• [17] M. Paluszyński, H. Šikić, G. Weiss, S.-L. Xiao , Generalized low pass filters and MRA frame wavelets, J. Geom. Anal., 11 (2001), 311–342.

• [18] The Wutam Consortium , Basic properties of wavelets, J. Fourier Anal. Appl., 4 (1998), 575–594.