Mini-Workshop: Wavelets and Frames
Authors: Feichtinger Hans Georg , Jorgensen Palle E.T. , Larson Dave , Olafsson Gestur 
Published: December 2009

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}}

:

\tag{1}
f = \sum_{n \in \mathbb{Z}^{2d}} \langle f, \tilde g_n \rangle g_n, \text{ for all } f \in \mathcal{H}.

In particular,

\mathcal{G}

is a frame if (and only if) the so called
frame operator

Sf = \sum_{n \in \mathbb{Z}^{2d}}\langle f, g_n \rangle g_n,

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

S \tilde{\mathcal{G}}= \mathcal{G}.

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

*

-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
and
translations
: 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: