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Interpolation operators associated with sub-frame sets

Author: Deguang Han
Journal: Proc. Amer. Math. Soc. 131 (2003), 275-284
MSC (2000): Primary 42C15, 47B38
Published electronically: June 3, 2002
MathSciNet review: 1929047
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Abstract: Interpolation operators associated with wavelets sets introduced by Dai and Larson play an important role in their operator algebraic approach to wavelet theory. These operators are also related to the von Neumann subalgebras in the ``local commutant'' space, which provides the parametrizations of wavelets. It is a particularly interesting question of how to construct operators which are in the local commutant but not in the commutant. Motivated by some questions about interpolation family and C*-algebras in the local commutant, we investigate the interpolation partial isometry operators induced by sub-frame sets. In particular we introduce the $2\pi $-congruence domain of the associated mapping between two sub-frame sets and use it to characterize these partial isometries in the local commutant. As an application, we obtain that if two wavelet sets have the same $2\pi $-congruence domain, then one is a multiresolution analysis (MRA) wavelet set which implies that the other is also an MRA wavelet set.

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  • [DL] X. Dai and D. Larson, Wandering vectors for unitary systems and orthogonal wavelets, Memoirs Amer. Math. Soc., 640 (1998). MR 98m:47067
  • [DLS] X. Dai, D. R. Larson and D. Speegle, Wavelets in $\mathbb{R} ^{n}$, J. Fourier Analysis and Applications, 3 (1997), 451-456. MR 98m:42048
  • [DLu] X. Dai and S. Lu, Wavelet in subspaces, Michigan J. Math., 43 (1996), 81-98. MR 97m:42021
  • [DDG] X. Dai, Y. Diao and Q. Gu, Subspaces with normalized tight frames in $\mathbb{R} $, Proc. Amer. Math. Soc., 129 (2001), 2045-2055. MR 2002a:46006
  • [DDGH] X. Dai, Y. Diao, Q. Gu and D. Han, Frame wavelet sets in $\mathbb{R} ^{d}$, Proc. Amer. Math. Soc., to appear.
  • [Dau] I. Daubechies, Ten Lectures on Wavelets, CBS-NSF Regional Conferences in Applied Mathematics, 61, SIAM (1992). MR 93e:42045
  • [Han] D. Han, Unitary systems, wavelets and operator algebras, Ph.D. thesis, Texas A&M University, 1998.
  • [HL] D. Han and D. R. Larson, Frames, bases and group representations, Memoirs Amer. Math. Soc., 697 (2000). MR 2001a:47013
  • [HW] E. Hernandez and G. Weiss, A First Course in Wavelets, Studies in Advanced Mathematics, CRC Press, 1996. MR 97i:42015
  • [La] D. Larson, Von Neumann algebras and wavelets. Operator algebras and applications (Samos, 1996), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 495, Kluwer Acad. Publ., Dordrecht (1997), 267-312. MR 98g:46091
  • [Ma] S. Mallat, Multiresolution approximations and wavelet orthonormal basis of $ L ^2 (\mathbb{R} )$, Trans. Amer. Math. Soc., 315 (1989), 69-87. MR 90e:42046
  • [Me] Y. Meyer, Ondelettes et Operateurs, Herman, Paris, 1990. MR 93i:42002; MR 93i:42003
  • [Sp] D. Speegle, The s-elementary wavelets are connected, Proc. Amer. Math. Soc. 127(1999), 223-233. MR 99b:42045

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Additional Information

Deguang Han
Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816

Keywords: Wavelet, sub-frame set, interpolation operators, congruence domain, multiresolution analysis, MRA wavelet set
Received by editor(s): February 1, 2001
Received by editor(s) in revised form: September 10, 2001
Published electronically: June 3, 2002
Communicated by: David R. Larson
Article copyright: © Copyright 2002 American Mathematical Society

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