Mini-Workshop: Wavelets and Frames

Abstract

The workshop was centered around two important topics in modern harmonic analysis: “Wavelets and frames”, as well as the related topics “time-frequency analysis” and “operator algebras”.

The theory of frames, or stable redundant non-orthogonal expansions in Hilbert spaces, introduced by Duffin and Schaeffer in 1952, plays an important role in wavelet theory as well as in Gabor (time-frequency) analysis for functions in $L^2({\mathbb{R}}^d)$. Besides traditional and relevant applications of frames in signal processing, image processing, data compression, pattern matching, sampling theory, communication and data transmission, recently the use of frames also in numerical analysis for the solution of operator equation by adaptive schemes is investigated. These important applications motivated the study of frames as decompositions in classical Banach spaces, e.g. Lebesgue, Sobolev, Besov, and modulation spaces. Fundamental concepts on operator theory, as well as on the theory of representations of groups and algebras are also involved and they have inspired new directions within frame theory with applications in pseudodifferential operator and symbolic calculus and mathematical physics.

Any element $f$ of the Hilbert space $\mathcal{H}$ can be expanded as a series with respect to a frame $\mathcal{G}=\{g_n{\}}_{n\in {\mathbb{Z}}^{2d}}$ in $\mathcal{H}$, and the coefficients of such expansion can be computed as scalar products of $f$ with respect to a dual frame $\tilde{\mathcal{G}}=\{{\tilde{g}}_n{\}}_{n\in {\mathbb{Z}}^{2d}}$:

In particular, $\mathcal{G}$ is a frame if (and only if) the so called frame operator

is continuous and continuously invertible on its range. Then there exists a canonical choice of a possible dual frame (delivering the minimal norm coefficient) defined by the equation

The existence of a dual frame makes the expansion (1) work. On the other hand, it may be a hard problem to predict properties of the canonical dual frame since it is only implicitly defined by the previous equation, and not always is there an efficient way of computation approximations at hand. This motivated the so called localization theory for frames, making use of well-chosen Banach $\ast$-algebras of infinite matrices. They allow to deduce relevant properties of the canonical dual and to extend the Hilbert space concept of frames to Banach frames which characterize corresponding families of Banach spaces.

Another problem within frame theory concerns structured families of functions, depending perhaps on several parameters, and the question of whether such a family constitutes a frame for $L^2({\mathbb{R}}^d)$. Classical examples are the following ones. Gabor frames are frames in $L^2({\mathbb{R}}^d)$ constructed by modulations and translations: given a square-integrable function $g$ our sequence is $g_{nm}(x)=e^{2\pi i(m,x)}g(x-n)$, $(n,m)\in \Lambda$, where $\Lambda$ is a discrete subset of ${\mathbb{R}}^{2d}$. The wavelet frames are constructed using dilations andtranslations: given a set $\Delta \subset \mathrm{GL}(d,\mathbb{R})$ and $\Gamma \subset {\mathbb{R}}^d$, as well as a suitable square integrable function $\psi$, we set ${\psi }_{D,\gamma }(x)=|\det D{|}^{1/2}\psi (Dx+\gamma )$ for $D\in \Delta$ and $\gamma \in \Gamma$. They are canonically related to Besov spaces. The reader can find several interesting questions and problems related to those concepts in the following abstracts.

We would like to exemplify here two simple existence problems. If the density of the points in $\Lambda$ is too small, then a Gabor frame cannot be constructed, and if the density is too large, then one can construct a frame, but not a basis. Suitable definitions of density and their relations with respect to the existence of frames in one of the current relevant topics in the frame theory.

In the wavelet case, an interesting problem has been the construction of wavelet sets. Given the set $\Delta$ and $\Lambda$, find the measurable subsets $\Omega \subset {\mathbb{R}}^d$ of positive, and finite measure, such that, with $\hat{\psi }={\chi }_{\Omega }$, the sequence $\{{\psi }_{D,\gamma }{\}}_{D\in \Delta ,\gamma \in \Gamma }$, is an orthogonal basis for $L^2({\mathbb{R}}^d)$. Such a set is called a wavelet set. This line of work includes both geometry (tilings of ${\mathbb{R}}^d$) and analysis (the Fuglede conjecture). More general question is when $\{{\psi }_{D,\gamma }{\}}_{D\in \Delta ,\gamma \in \Gamma }$ can be a frame.

Other more general frames, called wave packets, can be constructed as combinations of modulations, translations and dilations to interpolate the time-frequency properties of analysis of Gabor and wavelet frames. Interesting problems related to density and existence of such frames are an important direction of research and connections with new Banach spaces (for example $\alpha$-modulation spaces), Lie groups (for example the affine Weyl–Heisenberg group), and representation theory (for example the Stone–Von Neumann representation) are currently fruitful fields of investigation. All these families of frames are generated by the common action of translations. Shift invariant spaces and their generators constituted the main building blocks from which to start the construction of more complicated systems. They showed relevant uses in engineering, signal and image processing, being one of the most prominent branch in the applications.

Rather than formal presentations of recent advances in the field, this workshop tried instead to aim at outlining the important problems and directions, as we see it, for future research, and to discuss the impact of the current main trends. In particular, the talks were often informal with weight on interaction between the speaker and the audience, both in form of discussion and general comments. A special problem session was organized by D. Larson one afternoon. Another afternoon session was devoted to talks and informal discussions of further open problems, new directions, and trends.

The topics that emerged in these discussions included the following general areas:

1. Functional equations and approximation theory: wavelet approximation in numerical analysis, PDE, and mathematical physics. At the meeting, we discussed some operator theoretic methods that resonate with what numerical analysts want, and questions about localizing wavelets. We refer to the abstracts by M. Frank and K. Urban for more details. Two workshop lectures covered connections to numerical analysis and PDE.
2. Gabor frames: We had many discussions, much activity, and several talks on aspects of this. H. Feichtinger explained some important results and discussed some open problems involving frames and Gelfand triples. K. Gröchenig gave a lecture on new formulations and results generalizing Wiener's inversion theorems, in particular for twisted convolution algebras and Gabor frames. The applications are striking in that they yield sharper frame bounds. And they involve non-commutative geometry and other operator algebraic tools. C. Heil discussed the basic properties of frames which are not bases, and in particular he discussed the current status of the still-open conjecture that every finite subset of a Gabor frame is linearly independent. A related problem is that there do not exist any explicit estimates of the frame bounds of finite sets of time-frequency shifts.
3. Continuous vs. discrete wavelet transforms: We had several talks at the Oberwolfach workshop where the various operations, translation, scaling, phase modulation, and rotation, get incorporated into a single group. H. Führ and G. Ólafsson gave talks, where links to Lie groups and their representations were discussed. This viewpoint seems to hold promise for new directions, and for unifying a number of current wavelet constructions, tomography, scale-angle representations, parabolic scaling, wavelet packets, curvelets, ridgelets, de-noising$\dots$ Wavelets are usually thought of as frames in function spaces constructed by translations and dilations. Much less is understood in the case of compact manifolds such as the $n$-dimensional sphere, where both “dilations” and “translations” are not obviously defined. The talk by Ilgewska–Nowak explained some of her joint work with M. Holschneider on the construction of discrete wavelet transforms on the sphere.
4. Harmonic analysis of Iterated Function Systems (IFS): Several of the participants have worked on problems in the area, and P. Jorgensen spoke about past work, and directions for the future. The iterated function systems he discussed are closely related to the study of spectral pairs and the Fuglede problem. Recent work by Terence Tao makes the subject especially current.
5. Multiplicity theory, spectral functions, grammians, generators for translation invariant subspaces, and approximation rates: We had joint activity at the workshop on problems in the general area, and we anticipate joint papers emerging from it. A. Aldroubi lectured on the engineering motivations. In particular he discussed translation invariant subspaces of $L^2(\mathbb{R})$ where two lattice-scales are involved, and issues about localizing the corresponding generating functions for such subspaces. O. Christensen presented an equivalent condition for two functions generating dual frame pairs via translation. The result lead to a way of finding a dual of a given frame, which belongs to a prescribed subspace. Several open questions related to this were discussed.
6. Decompositions of operators and construction of frames: D. Larson discussed the problem of when is a positive operator a sum of finitely many orthogonal projections, and related it to frame theory. Problems and some recent results and techniques of D. Larson and K. Kornelson were discussed in this context, involving other related types of targeted decompositions of operators. In response, H. Feichtinger and K. Gröchenig pointed out that similar techniques just may lead to progress on a certain problem in modulation space theory. There are plans to follow up on this lead.
7. Wave packets: We had two talks at the workshop about this broad research area. G. Kutyniok gave a talk about the role of the geometric structure of sets of parameters of wave packets for the functional properties of associated systems of functions. In this context some recent results of D. Speegle, G. Kutyniok, and W. Czaja were discussed. M. Fornasier presented the construction of a specific family of wave packet frames for $L^2(\mathbb{R})$ depending on a parameter $\alpha \in [0,1)$, as a mixing tuner between Gabor and wavelet frames. These more classical and well-known frames arise as special and extreme cases.