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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.

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

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