FRG: Collaborative Research: Focused Research on Wavelets, Frames, and Operator Theory

NSF Focused Research Group:

Wavelets, Frames, and Operator Theory

This FRG will enhance collaboration among the members by:

holding regular think-tank-style workshops,
involving students and postdocs in those workshops,
exchanging students for short visits among the Principal Investigators,
forming project subgroups to attack specific problem areas.

Contents
   A sample of the FRG research problems

   Project areas

   Participants

   Events (upcoming events in red)

   Reports

A sample of the FRG research problems:

Definition: A subset

is said to be a wavelet set for an expansive integral

$d\times d$ matrix

if the inverse Fourier transform

$\psi$ of

is a wavelet, i.e., if the double indexed family

$j\in \QTR{Bbb}{Z}$ ,

, is an orthonormal basis for

. (This is equivalent to

$\Omega$ tiling

$\QTR{Bbb}{R}^{d}$ both under the translations

and the scalings

Problem A: Wavelet set implementation. Because of the structure of wavelet sets, their explicit constructions, and the potential usefulness of dealing with only one wavelet in higher dimensions, there are two immediate and natural problems. We wish to develop recursive programmable schemes for applications, cf. [BT93, Stro00a, Wic94], as well as a theoretical formulation in terms of frames, cf. [HL00, Han97].

This latter approach would sacrifice orthonormality, which is often not a sacrifice in applications dealing with noise reduction and stable representation. On the other hand, the frame theoretic approach would reduce the recursive complexity, it would add smoothness in the spectral domain, and it would provide an iterative method of signal reconstruction. Related to this whole issue is the necessity of a critical comparison of the geometrical constructions in [BS01] and the theoretical but constructive approach in [BMM99] in terms of multiplicity functions and finite von Neumann algebras.

Further, the examples of wavelet sets are typically fractal-like, in the sense that the sets are given as limits of infinite processes. Numerically, this could pose difficulties. Consequently, one may seek wavelet sets that are, for example, finite unions of rectangles. In this regard, Merrill has discovered a family of multiwavelets (with multiplicity 2 in 2 dimensions) of this type, cf. the geometric examples after the complete iteration in [DL98].

[BMM99] L.W. Baggett, H. A. Medina, and K. D. Merrill, Generalized multi-resolution analysis, and a construction procedure for all wavelet sets in Rⁿ, J. Fourier Anal. Appl., 5 (1999), 563-573.

[BS01] J. J. Benedetto and S. Sumetkijakan, A fractal set constructed from a class of wavelet sets, Inverse problems, image analysis, and medical imaging (New Orleans, LA, 2001) (M. Zuhair Nashed and Otmar Scherzer, eds.), Contemporary Mathematics, vol. 313, American Mathematical Society, Providence, RI, 2002, pp. 19-35.

[BT93] J. J. Benedetto and A. Teolis, A wavelet auditory model and data compression, Appl. Comput. Harmon. Analy., 1 (1993), 3-28.

[DL98] X. Dai and D. R. Larson, Wandering vectors for unitary systems and orthogonal wavelets, Mem. Amer. Math. Soc., 134 (1998), no. 640.

[Han97] B. Han, On dual wavelet tight frames, Appl. Comput. Harmon. Anal., 4 (1997), 380-413.

[HL00] D. Han and D. R. Larson, Frames, bases and group representations, Mem. Amer. Math. Soc., 147 (2000), no. 697.

[Stro00a] T. Strohmer, Numerical analysis of the non-uniform sampling problem, J. Comput. Appl. Math., 122 (2000), 297-316.

[Wic94] M. V. Wickerhauser, Adapted Wavelet Analysis from Theory to Software, A K Peters Ltd., Wellesley, MA, 1994.
Definition: (see [JoPe92]) A measurable set $\Omega$ in $\QTR{Bbb}{R}^{d}$ , of finite positive measure, is said to be a spectral set if has an orthogonal basis of real exponentials for a subset $\Lambda$ of points . If an orthogonal basis exists, $\Lambda$ is called a spectrum. $\Omega$ is said to be a tiling if it can be made to tile $\QTR{Bbb}{R}^{d}$ using only translations.
[JoPe92] P. E. T. Jorgensen and S. Pedersen, Spectral theory for Borel sets in $\QTR{Bbb}{R}^{n}$ of finite measure, J. Funct. Anal., 107 (1992), 72-104.
Problem B: Spectra-Tilings Conjecture. Let with . If $\Omega$ is a spectral set, then $\Omega$ tiles $\QTR{Bbb}{R}^{d}$ by translation. Further, there is a one-to-one correspondence between the spectra of $\Omega$ and the set of tilings of $\Omega$ .

The Fuglede conjecture, in its original primitive form, is 40 years old, but it has come to signify a much more general principle: It is now used to describe any one of several correspondences between spectrum and geometry, much in the spirit of Mark Kac's "Can you hear the shape of a drum?" theme. In its original form, the Fuglede conjecture states that an open set S of finite positive measure in R^d admits a Fourier basis F if and only if it tiles R^d with translations. This form of the conjecture is still open, but it is recognized as being difficult, involving combinatorics and number theory. A number of serious attempts at counterexamples have not succeeded. But in many applications, more detailed information is available about the context of the problem: If for example S is known to be the finite union of non-overlapping intervals, then the possible spectral pairs (S,F) are of significance in combinatorics and in sampling problems. Therefore much effort in the FRG has been focused on cases when S is already known to satisfy both the basis condition and the tile condition. Then the question is the correspondence between spectrum and tile-translations, or more generally between spectrum and geometry. This is "the generalized Fuglede conjecture", and it has been extraordinarily fruitful. The simplest case arises when the vectors that produce a Fourier basis are also translation vectors in a tiling. Several of us in the FRG have published papers on this: For example, if S is the d-cube, members in the FRG and their co-workers showed that the frequencies F in a Fourier basis and the vectors V that make S tile coincide, i.e., F = V, and when d = 3, we found the possibilities for F. Equivalently, (S,F) is a spectral pair if and only if (S,F) is a tile system. But spectral/tile duality is a central theme in many wavelet problems, in iterated function systems, in spectral theory for Hilbert spaces built from Hausdorff measure H^s, of fractional dimension s. They are central in harmonic analysis, see for example the websites of Terence Tao and I. Laba, and this FRG website. In its general form, the Fuglede conjecture is central to how we construct wavelet bases, and Gabor frames. In the FRG, Jorgensen and his Ph.D. student D. Dutkay proved that the spectral/tile principle applies to H^s, Hausdorff measure, 0 < s < 1, and that the duality principle lets us write down wavelet bases in H^s-Hilbert spaces. Even for the middle-third Cantor set, this is new and significant. The middle-third Cantor set is one of the fractals which does not have a Fourier basis. Many others do! (A sample of researchers connected to the FRG and with papers on the subject: Palle Jorgensen, Steen Pedersen, Yang Wang, Jeff Lagarias, Izabella Laba, Alex Iosevich, Terence Tao,... A sample of papers as of May 15, 2003: [B1]-[B14] below.) Research on exact harmonic analysis-duality principles in geometric measure theory are still in their infancy. But there is a lot of interest in this. Note that R. Strichartz's NSF-funded undergraduate research projects have this focus. We plan to continue to work with Strichartz and his team in this direction.
This circle of problems around the Fuglede conjecture is not isolated. Don't think of math as finite. Rather: "Infinite in all directions", is more like it! We usually see that each answered question opens up three new questions. That is especially true for the good problems in math. (Hmm...!, it would sound like math is not efficient, but the good stuff is what we learn from solutions to problems! Perhaps more so than the problems themselves.) Anyway what we call "The Fuglede problem", or conjecture (I forgot when it was elevated to a conjecture!) is in fact many problems..! A major theme in math is that of understanding connections between geometry (for example tiling) and some kind of spectrum. What I learned from the last wonderful Terence Tao paper, and papers of others, is that it may be possible to do some kind of functor from geometry to an "easier" setup (finite combinatorics and algebra) involving only certain finite cyclic groups G. When is this is successful? If the assignment is a functor! We have to be able to get an implication like this: Within the spectral setting coming from a suitable Hadamard matrix, verify the implication: "No solution in the finite case (a particular G^d for some d)" implies "no solution to the original problem!" The original problem here is the tiling question. Looked at this way, we are just scratching the tip of an ice-berg.
[B1] D. Dutkay, P. Jorgensen, Wavelets on fractals, preprint June 2003, Univ. of Iowa, submitted to Revista Matemática Iberoamericana.
[B2] B. Fuglede, Commuting self-adjoint partial differential operators and a group-theoretic problem, J. Funct. Anal. 16 (1974), 101-121.
[B3] A. Granville, I. Laba, Y. Wang, On finite sets which tile the integers, Preprint Dec 19, 2001.
[B4] A. Iosevich, N. Katz, T. Tao, Convex bodies with a point of curvature do not have Fourier bases, Amer. J. Math. 123 (2001), 115-120.
[B5] A. Iosevich, S. Pedersen, Spectral and tiling properties of the unit cube, Internat. Math. Res. Notices 1998 (1998), no. 16, 819-828.
[B6] P. Jorgensen, Spectral theory of finite-volume domains in Rⁿ, Adv. in Math. 44 (1982), 105-120.
[B7] P. Jorgensen, S. Pedersen, Spectral theory for Borel sets in Rⁿ of finite measure, J. Funct Anal. 107 (1992), 72-104.
[B8] P. Jorgensen, S. Pedersen, Spectral pairs in Cartesian coordinates, J. Fourier Anal. Appl. 5 (1999), 285-302.
[B9] I. Laba, Fuglede's conjecture for a union of two intervals, Proc. Amer. Math. Soc. 129 (2001), 2965--2972.
[B10] J.C. Lagarias, J.A. Reeds, Y. Wang, Orthonormal bases of exponentials for the n-cube, Duke Math. J. 103 (2000), 25-37.
[B11] J.C. Lagarias, Y. Wang, Tiling the line with translates of one tile, Invent. Math. 124 (1996), 341-365.
[B12] S. Pedersen, Y. Wang, Universal spectra, universal tiling sets and the spectral set conjecture, Math. Scand. 88 (2001), 246--256.
[B13] Y. Wang, Wavelets, tiling, and spectral sets, Duke Math. J. 114 (2002), 43--57.
[B14] Y. Liu and Y. Wang, The uniformity of non-uniform Gabor bases, Adv. Comput. Math. 18 (2003), no. 2--4 (special issue on Frames edited by A. Aldroubi and Q. Sun), 345--355.
Problem C: Irregular Translations. The result of Wang [Wan01] on admissible translation and dilation sets for wavelets arising from wavelet sets shows that these sets can be irregular. The same has not been studied for other type of wavelets. What if we change the wavelet set condition to Meyer wavelets? What about compactly supported wavelets and MRA wavelets? Wang's approach differs substantially from the approaches in [LWWW02] and [Füh01]. Could it be used to study the admissible dilation sets in a continuous wavelet transform? How could an multiresolution analysis be defined with more than one scaling factor? If so, will there be potential for new applications? The members of the FRG have much combined expertise in areas related to wavelets, wavelet sets, and irregular sampling. In particular, Larson was the first to study wavelet sets, Aldroubi, Benedetto and Heil have extensive expertise in irregular sampling and MRA wavelets, and Jorgensen and Ólafsson have studied wavelets and wavelet sets from the point of view of representation theory, which may offer another important insight into this problem.

[Füh01] H. Führ, Admissible vectors for the regular representation, Proc. Amer. Math. Soc. 130 (2002), no. 10, 2959-2970 (electronic).

[LWWW02] R. S. Laugesen, N. Weaver, G. Weiss and E. Wilson, A characterization of the higher dimensional groups associated with continuous wavelets, J. Geom. Anal. 12 (2002), no. 1, 89-102.

[Wan01] Y. Wang, Wavelets, tiling and spectral sets, Duke Math. J. 114 (2002), no. 1, 43-57.

Project areas:

Wavelet sets: implementation and abstract wavelet theory (manager: L.W. Baggett).

Finite, Fourier, and Gabor frames: sampling and Beurling-Landau theory (manager: C.E. Heil).

Spectral-tile duality in wavelet theory (manager: P.E.T. Jorgensen); for details see the web site of Izabella Laba, especially the section Tilings of Rⁿ and the spectral set conjecture.

Work displayed on this page was supported in part by the U.S. National Science Foundation under grants DMS-9987777 and DMS-0139473(FRG).

This page was last modified on 2 Apr 2004 by Brian Treadway.

to Palle Jorgensen's home page

Wavelets, Frames, and Operator Theory

This FRG will enhance collaboration among the members by:

Contents A sample of the FRG research problems Project areas Participants Events (upcoming events in red) Reports

A sample of the FRG research problems:

Project areas:

Participants:

Events (upcoming events in red):

Reports:

Contents
A sample of the FRG research problems

Project areas

Participants

Events (upcoming events in red)

Reports