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References (33)
-
M. Dobrescu (2005)
Wavelet sets with and without groups and multiresolution analysis -
The many faces of wavelet sets, history and open problems -
N. Lord, Nicolas Bourbaki (1989)
Lie Groups and Lie Algebras: Chapters 1-3 -
J. Humphreys (1990)
Reflection Groups and Coxeter Groups -
P. Gunnells (2006)
Notices of the AMS -
R. Steinberg (1959)
FINITE REFLECTION GROUPSTransactions of the American Mathematical Society, 91
-
Technische Universität München, Germany E-mail address: massopust@ma.tum -
B. Fuglede (1974)
Commuting self-adjoint partial differential operators and a group theoretic problemJournal of Functional Analysis, 16
-
N. Bourbaki (1998)
Lie groups and Lie algebras -
Weaver Laugesen, Weiss, Wilson
WAVELETS , WAVELET SETS , AND LINEAR ACTIONS ON -
D. Larson, P. Massopust (2007)
Coxeter Groups and Wavelet SetsarXiv: Functional Analysis
-
Yang Wang (2002)
Wavelets, tiling, and spectral setsDuke Mathematical Journal, 114
-
J. Lagarias, Yang Wang (1997)
Spectral Sets and Factorizations of Finite Abelian GroupsJournal of Functional Analysis, 145
-
X. Dai, D. Larson, D. Speegle (1997)
Wavelet sets in ℝnJournal of Fourier Analysis and Applications, 3
-
(2007)
Wavelet-Mengen und Spiegelungsgruppen -
Michael Hoffman, W. Withers (1988)
Generalized Chebyshev polynomials associated with affine Weyl groupsTransactions of the American Mathematical Society, 308
-
M. Kolountzakis, M. Matolcsi (2004)
Tiles with no spectra, 18
-
G. Ólafsson, D. Speegle (2004)
Wavelets, frames and operator theory -
(1997)
Wavelet sets in R n . II. In Wavelets, multiwavelets , and their applications -
T. Tao (2003)
Fuglede's conjecture is false in 5 and higher dimensionsMathematical Research Letters, 11
-
C. Benson, L. Grove (1985)
Finite Reflection Groups -
J. Humphreys (1990)
Reflection groups and Coxeter groups: Coxeter groups -
A. Iosevich, N. Katz, T. Tao (1999)
Convex bodies with a point of curvature do not have Fourier basesAmerican Journal of Mathematics, 123
-
Icrp (2009)
Chapters 4–6Annals of the ICRP, 39
-
M. Dobrescu, G. Ólafsson (2007)
Wavelet sets without groupsarXiv: Functional Analysis
-
Peter Cameron (2007)
Regular polytopes -
M. Dobrescu, G. Ólafsson (2007)
Coxeter Groups, Wavelets, Multiresolution and SamplingarXiv: Functional Analysis
-
(2004)
Speegle: Wavelets -
P. Jorgensen, S. Pedersen (1998)
Orthogonal harmonic analysis of fractal measuresElectronic Research Announcements of The American Mathematical Society, 4
-
(1998)
and D -
X. Dai, D. Larson (1998)
Wandering Vectors for Unitary Systems and Orthogonal Wavelets -
X. Dai, D.R. Larson, D.M. Speegle (1998)
Wavelets, Multiwavelets, and Their Applications -
(2006)
Cells in Coxeter Groups, in Notices of the AMS
- Publisher
- Springer Journals
- Copyright
- Copyright © 2009 by Springer Science+Business Media B.V.
- Subject
- Mathematics; Mechanics; Statistical Physics, Dynamical Systems and Complexity; Theoretical, Mathematical and Computational Physics; Computer Science, general; Mathematics, general
- ISSN
- 0167-8019
- eISSN
- 1572-9036
- DOI
- 10.1007/s10440-008-9424-y
- Publisher site
- See Article on Publisher Site
Abstract
In this article we show that there exist measurable sets W⊂ℝ2 with finite measure that tile ℝ2 in a measurable way under the action of a expansive matrix A, an affine Weyl group $\widetilde{W}$ , and a full rank lattice $\widetilde{\varGamma}\subset\mathbb{R}^{2}$ . This note is follow-up research to the earlier article “Coxeter groups and wavelet sets” by the first and second authors, and is also relevant to the earlier article “Coxeter groups, wavelets, multiresolution and sampling” by M. Dobrescu and the third author. After writing these two articles, the three authors participated in a workshop at the Banff Center on “Operator methods in fractal analysis, wavelets and dynamical systems,” December 2–7, 2006, organized by O. Bratteli, P. Jorgensen, D. Kribs, G. Ólafsson, and S. Silvestrov, and discussed the interrelationships and differences between the articles, and worked on two open problems posed in the Larson-Massopust article. We solved part of Problem 2, including a surprising positive solution to a conjecture that was raised, and we present our results in this article.
Journal
Acta Applicandae Mathematicae – Springer Journals
Published: Jan 14, 2009