Interpolation operators associated with sub-frame sets
HTML articles powered by AMS MathViewer
- by Deguang Han PDF
- Proc. Amer. Math. Soc. 131 (2003), 275-284 Request permission
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.References
- Xingde Dai and David R. Larson, Wandering vectors for unitary systems and orthogonal wavelets, Mem. Amer. Math. Soc. 134 (1998), no. 640, viii+68. MR 1432142, DOI 10.1090/memo/0640
- Xingde Dai, David R. Larson, and Darrin M. Speegle, Wavelet sets in $\mathbf R^n$, J. Fourier Anal. Appl. 3 (1997), no. 4, 451–456. MR 1468374, DOI 10.1007/BF02649106
- Xingde Dai and Shijie Lu, Wavelets in subspaces, Michigan Math. J. 43 (1996), no. 1, 81–98. MR 1381601, DOI 10.1307/mmj/1029005391
- X. Dai, Y. Diao, and Q. Gu, Frame wavelet sets in $\Bbb R$, Proc. Amer. Math. Soc. 129 (2001), no. 7, 2045–2055. MR 1825916, DOI 10.1090/S0002-9939-00-05873-1
- X. Dai, Y. Diao, Q. Gu and D. Han, Frame wavelet sets in $\mathbb {R}^{d}$, Proc. Amer. Math. Soc., to appear.
- Ingrid Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1162107, DOI 10.1137/1.9781611970104
- D. Han, Unitary systems, wavelets and operator algebras, Ph.D. thesis, Texas A&M University, 1998.
- Deguang Han and David R. Larson, Frames, bases and group representations, Mem. Amer. Math. Soc. 147 (2000), no. 697, x+94. MR 1686653, DOI 10.1090/memo/0697
- Eugenio Hernández and Guido Weiss, A first course on wavelets, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1996. With a foreword by Yves Meyer. MR 1408902, DOI 10.1201/9781420049985
- D. R. Larson, Von Neumann algebras and wavelets, Operator algebras and applications (Samos, 1996) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 495, Kluwer Acad. Publ., Dordrecht, 1997, pp. 267–312. MR 1462685
- Stephane G. Mallat, Multiresolution approximations and wavelet orthonormal bases of $L^2(\textbf {R})$, Trans. Amer. Math. Soc. 315 (1989), no. 1, 69–87. MR 1008470, DOI 10.1090/S0002-9947-1989-1008470-5
- Erich Rothe, Topological proofs of uniqueness theorems in the theory of differential and integral equations, Bull. Amer. Math. Soc. 45 (1939), 606–613. MR 93, DOI 10.1090/S0002-9904-1939-07048-1
- D. M. Speegle, The $s$-elementary wavelets are path-connected, Proc. Amer. Math. Soc. 127 (1999), no. 1, 223–233. MR 1468204, DOI 10.1090/S0002-9939-99-04555-4
Additional Information
- Deguang Han
- Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816
- Email: dhan@pegasus.cc.ucf.edu
- 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
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 275-284
- MSC (2000): Primary 42C15, 47B38
- DOI: https://doi.org/10.1090/S0002-9939-02-06658-3
- MathSciNet review: 1929047