The theory of matrix-valued multiresolution analysis frames

Authors

1 School of Science‎, ‎Beijing Jiaotong University‎, ‎Beijing‎, ‎100044‎, ‎China.

2 Faculty of Mathematics Science‎ , ‎Tianjin normal‎ ‎University‎, ‎Tianjin‎, ‎300074‎, ‎China

Abstract

‎A generalization of matrix-valued multiresolution analysis (MMRA)‎ ‎to matrix-valued frames‎, ‎and the constructions of matrix-valued‎ ‎frames are considered and characterized‎. ‎A matrix-valued frame‎ ‎multiresolution analysis is defined in this paper‎. ‎We provide‎ ‎necessary and sufficient conditions for constructing matrix-valued‎ ‎frames and Riesz bases of translates‎, ‎and give the calculation‎ ‎method of matrix-valued dual Riesz basis‎. ‎These conclusions are‎ ‎useful in providing theoretical basis for constructing‎ ‎matrix-valued frames and Riesz basis.

Keywords


A. S. Antolin and R. A. Zalik, Matrix-valued wavelets and multiresolution analysis, J. Appl. Funct. Anal. 7 (2012), no. 1-2, 13--25.
J. J. Benedetto and S. Li, The theory of multiresolution analysis frames and applications to filter banks, Appl. Comput. Harmon. Anal. 5 (1998), no. 4, 389--427.
Q. Chen, Z. X. Cheng and C. L. Wang, Existence and construction of compactly supported biorthogonal multiple vector-valued wavelets, J. Appl. Math. Comput. 22 (2006), no. 3, 101--115.
Q. Chen and Z. Shi, Construction and properties of orthogonal matrix valued waveletsand wavelet packets, Chaos, Solitons Fractals 37 (2008), no. 1 , 75--86.
L. Cui, B. Zhai and T. Zhang, Existence and design of biorthogonal matrix-valued wavelets, Nonlinear Anal. Real World Appl. 10 (2009), no. 5, 2679--2687.
J. Geronimo, D. P. Hardin and P. Massopust, Fractal functions and wavelet expansions based on several scaling functions, J. Approx. Theory 78 (1994), no. 3, 373--401.
P. Ginzberg and A. T. Walden, Matrix-valued and quaternion wavelets, IEEE Trans. Signal Process. 61 (2013), no. 6, 1357--1367.
Q. Jiang, On the design of multifilter banks and orthonormal multiwavelet bases, IEEE Trans. Signal Process. 46 (1998) 3292--3303.
W. Li and P. Zhao, A study on minimum-energy vector-valued wavelets tight frames, International Conference on Information Science and Technology (2012) 436--440.
A. Ron and Z. Shen, Frames and stable bases for shift-invariant subspaces of L2(Rd), Canad. J. Math. 47 (1995), no. 5, 1051--1094.
F. A. Shah and N. A. Sheikh, Construction of vector-valued multivariate wavelet frame packets, Thai J. Math. 10 (2012), no. 2, 401--414.
K. Slavakis and I. Yamada, Biorthogonal unconditional bases of compactly supported matrix valued wavelets, Numer. Funct. Anal. Optim. 22 (2001), no. 1-2, 223--253.
A. T. Walden and A. Serroukh, Wavelet analysis of matrix-valued time series, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 458 (2002), no. 2017, 157--179.
X. G. Xia and B.W. Suter, Vector-valued wavelets and vector filter banks, IEEE Trans. Signal Process, 44 (1996) 508--518.
X. G. Xia, Orthonormal matrix valued wavelets and matrix Karhunen-Love expansion. Wavelets, multiwavelets, and their applications (San Diego, CA, 1997), 159--175, Contemp. Math., 216, Amer. Math. Soc., Providence, 1998.
R. Young, An introduction to nonharmonic Fourier series, Academic Press, Inc., New York, 1980.
P. Zhao, C. Zhao and P. G. Casazza, Perturbation of regular sampling in shift-invariant spaces for frames, IEEE Trans. Inf. Theory 52 (2006), no. 10, 4643--4648.
P. Zhao, G. Liu and C. Zhao, A matrix-valued wavelet KL-like expansion for wide-sense stationary random processes, IEEE Trans. Signal Process. 52 (2004), no. 4, 914--920.