## Summer 2002 Program in Matrix Analysis and Wavelet Theory.

**Instructors:** Dave Larson with the
help of Ken Dykema, Keri Kornelson and Eric Weber.

This VIGRE seminar was offered in conjunction with our
NSF-funded Summer REU
program . To motivate wavelets, we begin with a few real-world
examples:

*Filtering.* A sound signal is often corrupted by noise (i.e.,
frequencies different from those in the desirable parts of the
signal). Signal analysis can be used to filter out this unwanted
noise. A Dolby filter, which filters out tape-hiss on cassette
tapes, is an example along these lines.

*Data Compression.* Digitized audio and video signals are
usually quite large, and are difficult to transmit electronically.
Efficient transmission of these signals often requires compression,
a process that eliminates the less significant parts of a signal.
Compression is used, for example, in transmitting fingerprints from
a police squad car to FBI Headquarters (in Washington DC) to
identify crime suspects.

*Detection.* Signals often have some feature that the user
wants to detect. For example, the sound made by a mechanical device
often changes when it does not operate correctly. A device that
detects this change would be useful to the machine operator.

Fourier analysis and wavelets are two of the basic tools used, in
signal analysis, to address the above issues.

*Fourier Analysis.* A Fourier series decomposes a signal
*f* into its trignonometric components, which vibrate at
various frequencies. The Fast Fourier transform (FFT) is an
efficient algorithm for calculating approximate values for the
Fourier coefficients. The Fourier coefficients can then be
manipulated according to the desired goal. If noise is to be
filtered-out, then the Fourier coefficients corresponding to the
unwanted frequencies can be eliminated. If the signal is to be
compressed, then the Fourier coefficients that are smaller (in
absolute value) than some specified tolerance can be discarded.
Problems in detection can be addressed by matching a subset of the
Fourier coefficients of *f* to a known profile of the type of
signal to be detected.

*Wavelets.* One disadvantage of Fourier series is that the
building blocks, sines and cosines, are periodic waves that
continue forever. While this approach may be quite appropriate for
filtering or compressing signals that have time-independent
wave-like features, other signals may have more localized features
that sines and cosines do not model very well. For example, suppose
an isolated noisy ``pop'' to a sound signal is to be filtered-out.
The graphs of sines and cosines do not resemble the pop's graph, an
isolated bump. A different set of building blocks, called
*wavelets*, are better suited to this type of signal. In a
rough sense, a wavelet resembles a wave that travels for one or
more periods and is nonzero only over a finite interval -- instead
of propagating forever as do sines and cosines.

A wavelet can be translated forward or backwards in time. It also
can be stretched or compressed, by scaling, to obtain low and high
frequency wavelets. Once a wavelet function is constructed, it can
be used to filter or compress signals in much the same manner as
Fourier series. A given signal is first expressed as a sum of
translations and scalings of the wavelet, and then the coefficients
corresponding to the unwanted terms are removed or modified.

Care must be taken in the construction of a wavelet to ensure that
its translates and rescalings satisfy orthogonality relationships
-- analogous to those of sines and cosines -- so that efficient
algorithms can be found for the computation of wavelet coefficients
of a given signal.

**Focus of this Program.** There is a fascinating interplay
between wavelets and operator theory (i.e. the theory of linear
maps between vector spaces). This interplay will be the key topic
of investigation during this summer program. We shall first
investigate the role of wandering vectors for unitary systems, both
in finite and infinite dimensions. Related topics include
introductory ideas in operator algebras, again both in finite and
infinite dimensions. In addition, we shall study minimally
supported frequency wavelets. This special class of wavelets has an
interesting internal structure, but also provides concrete examples
of important ideas that will be discussed. Finally, we will
investigate several interesting intrinsic problems dealing with
wavelet sets.

**Proposed Research Problems:**

- Open problems regarding wandering vectors for unitary systems
acting on R
^{n}or C^{n} - An open problem on the reflexivity of finite dimensional operator algebras which is purely algebraic in nature.
- Characterize interpolation pairs of minimally supported frequency wavelets.
- How can a wavelet set be perturbed to give rise to an interpolation pair?
- Does there exist a wavelet set in the support of the Fourier transform of any wavelet?