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VIGRE seminar, fall 2000: Problems in Matrix Analysis and Wavelet Theory

Instructors
David Larson, Eric Weber
Students enrolled
Scott Armstrong, Vincent Lemoine (undergraduate mathematics students); Scott Evans (undergraduate physics student); Eric Bahuaud, Troy Henderson, Ali-Amir Husain, Quoc Le Gia, Lovas Randrianarivony, Darren Rhea, Xiaofei Zhang (graduate mathematics students)
Description
The interplay between operator theory and wavelet theory has produced deep, rich results. We investigated 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 studied minimally supported frequency wavelets. This special class of wavelets has an interesting internal structure, but also provides concrete examples of important ideas that were discussed. Finally, we investigated several interesting intrinsic problems dealing with wavelet sets.
Examples of Research Problems
  1. Open problems regarding wandering vectors for unitary systems acting on Rn or Cn.
  2. An open problem on the reflexivity of finite dimensional operator algebras which is purely algebraic in nature.
  3. Characterize interpolation pairs of minimally supported frequency wavelets.
  4. How can a wavelet set be perturbed to give rise to an interpolation pair?
  5. Does there exist a wavelet set in the support of the Fourier transform of any wavelet?