Summer 2003 Program in Matrix Analysis and Wavelet Theory

Mentored by Dave Larson

Student Participants: from REU: Rachel Derber, Ivan Christov, James Hughes, Justin Turner, Phillip Watkins, Micah Hawkins, Matt Hirn, Bill Finkerkeller, Brady McCary, Andrew Felker; VIGRE participants: AURISPA, BENJAMIN BAUMGARTEN, ERIK MATTH DOSEV, DETELIN TODOROV HENDERSON, TROY LEE, I HITCHCOCK, JAMES MITCH KING, EMILY JEANNETTE MCCARY, BRADY NGUYEN, QUYNH NGA STRAWN, NATHANIEL KIRK TRENEV, DIMITAR VASILE WATKINS, PHILLIP DANIE ZHENG, BENTUO

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, a_n and b_n. 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.

Closely related to the orthonormal wavelets are the frame wavelets. Frame sequences have been used for many years by engineers for purposes of signal processing and data compression, in a manner much like the use of wavelet and other orthonormal bases. Frame wavelets are single vectors which generate frame sequences under the action of the wavelet unitary system.

Focus of this Program. There is a fascinating interplay between wavelets, frames, 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 investigate the role of wandering vectors for unitary systems, both in finite and infinite dimensions. Related topics include introductory ideas in frame theory, sampling theory, and operator algebras, again both in finite and infinite dimensions. Additional topics include wavelet sets and 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 Rⁿ or Cⁿ.
• Open problems concerning finite frames and their relationships with matrix analysis and operator theory.
• 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?
• An open problem in sampling theory.