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Local analysis of frame multiresolution analysis with a general dilation matrix

Published online by Cambridge University Press:  17 April 2009

Hong Oh Kim ,
Rae Young Kim  and
Jae Kun Lim
Hong Oh Kim
Affiliation:
Division of Applied Mathematics, KAIST, 373–1 Guseong-dong, Yuseong-gu, Daejeon 305–701, Korea e-mail: hkim@amath.kaist.ac.kr, rykim@amath.kaist.ac.kr
Rae Young Kim
Affiliation:
Division of Applied Mathematics, KAIST, 373–1 Guseong-dong, Yuseong-gu, Daejeon 305–701, Korea e-mail: hkim@amath.kaist.ac.kr, rykim@amath.kaist.ac.kr
Jae Kun Lim
Affiliation:
CHiPS KAIST, 373–1, Guseong-dong, Yuseong-gu, Daejeon, 305–701, Republic of Korea e-mail: jaekun@ftn.kaist.ac.kr
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Abstract

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A multivariate semi-orthogonal frame multiresolution analysis with a general integer dilation matrix and multiple scaling functions is considered. We first derive the formulas of the lengths of the inital (central) shift-invariant space V0 and the next dilation space V1, and, using these formulas, we then address the problem of the number of the elements of a wavelet set, that is, the length of the shift-invariant space W0 := V1 ⊖ V0. Finally, we show that there does not exist a ‘genuine’ frame multiresolution analysis for which V0 and V1 are quasi-stable spaces satisfying the usual length condition.

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 67 , Issue 2 , April 2003 , pp. 285 - 295
Copyright
Copyright © Australian Mathematical Society 2003

References

[1]Benedetto, J.J. and Li, S., ‘The theory of multiresolution analysis frames and applications to filter banks’, Appl. Comput. Harm. Anal. 5 (1998), 398–427.Google Scholar
[2]Benedetto, J.J. and Treiber, O.M., ‘Wavelet frames: multiresolutionanalysis and extension principle’, in Wavelet transforms and time-frequency signal analysis, (Debnath, L., Editor) (Birkhaüser, Boston, 2000), pp. 3–36.Google Scholar
[3]de Boor, C., DeVore, R. and Ron, A., ‘On the construction of multivariate (pre)wavelets’, Constr. Approx. 9 (1993), 123–166.CrossRefGoogle Scholar
[4]de Boor, C., DeVore, R. and Ron, A., ‘Approximations from shift-invariant subspaces of L 2(ℝd)’, Trans. Amer. Math. Soc. 341 (1994), 787–806.Google Scholar
[5]de Boor, C., DeVore, R. and Ron, A., ‘The structure of finitely generated shift-invariant spaces in L 2(ℝd)’, J. Funct. Anal. 119 (1994), 37–78.CrossRefGoogle Scholar
[6]Bownik, M., ‘The structure of shift-invariant subspaces of L 2(ℝn)’, J. Funct. Anal. 177 (2000), 282–309.Google Scholar
[7]Christensen, O., Deng, B. and Heil, C., ‘Density of Gabor frames’, Appl. Comput. Harmon. Anal. 7 (1999), 292–304.Google Scholar
[8]Gröchenig, K. and Madych, W. R., ‘Multiresolution analysis, Haar bases, and self-similar tilings of Rn’, IEEE Trans. Inform. Theory 38 (1992), 556–568.Google Scholar
[9]Heil, C. and Walnut, D., ‘Continuous and discrete wavelet transforms’, SIAM Rev. 31 (1989), 628–666.Google Scholar
[10]Helson, H., Lectures on invariant subspaces (Academic Press, New York, 1964).Google Scholar
[11]Jia, R.-Q., ‘Shift-invariant spaces and linear operator equations’, Israel J. Math. 103 (1998), 258–288.CrossRefGoogle Scholar
[12]Jia, R.-Q. and Micchelli, C.A., ‘Using the refinement equation for the construction of pre-wavelets, II: powers of two’, in Curves and Surfaces, (Laurent, P.J., Le Méhauté, A. and Schumaker, L., Editors) (Academic Press, New York), pp. 209–246.Google Scholar
[13]Kim, H.O. and Lim, J.K., ‘New characterizations of Riesz bases’, Appl. Comput. Harmon. Anal. 4 (1997), 222–229.Google Scholar
[14]Kim, H.O. and Lim, J.K., ‘On frame wavelets associated with frame multi-resolution analysis’, Appl. Comput. Harmon. Anal. 10 (2001), 61–70.CrossRefGoogle Scholar
[15]Kim, H.O. and Lim, J.K., ‘Applications of shift-invariant space theory to some problems of multi-resolution analysis of L 2(ℝd)’, in Wavelet Analysis and Applications, (Deng, D., Huang, D., Jia, R.-Q, Lin, W. and Wang, J., Editors) (American Mathematical Society/International Press, Boston, 2001), pp. 183–191.Google Scholar
[16]Lim, J.K., ‘Gramian analysis of multivariate frame multiresolution analyses’, Bull. Austral. Math. Soc. 66 (2002), 291–300.CrossRefGoogle Scholar
[17]Mallat, S., ‘Multiresolution approximations and wavelets’, Trans. Amer. Math. Soc. 315 (1989), 69–88.Google Scholar
[18]Meyer, Y., Wavelets and operators, Cambridge Studies in Advanced Mathematics 37 (Cambridge University Press, Cambridge, 1992).Google Scholar
[19]Ron, A. and Shen„, Z. ‘Frames and stable bases for shift-invariant subspaces of L 2(ℝd)’, Canad. J. Math. 47 (1995), 1051–1094.CrossRefGoogle Scholar
[20]Walters, P., An introduction to ergodic theory, Graduate Texts in Mathematics 79 (Springer-Verlag, New-York, Berlin, 1981).Google Scholar