|Date:||Fri, September 25, 2009|
|Place:||Research I Seminar Room|
Abstract: I will survey two case studies of nonlinear methods of approximating and representing a closed curve that are part of ongoing research projects (joint work with S. Harizanov (JUB), C. Madsen (Texas A&M), T. Shingel (now UCSD)). They are related to my core areas of expertise: approximation theory and multi-scale analysis.
The first case study is about matrix-valued periodic functions appearing in certain modeling areas (in Lie theory, they constitute the elements of the so called loop group associated with a Lie group), and pose and partially answer the classical problem of quantitative approximation by polynomials (the rate of approximation in Weierstrass' theorem).
The second case study is motivated by the question of how one can store a certain geometric object/manifold using very little space (the bit compression problem) under the assumption that it has a certain smoothness and low co-dimension. These are big words, what we are doing is analyze one recent approach called "normal multi-scale transform" for the simple case of a closed curve in R^2.
I will be slow, and explain the slides of a conference presentation in enough detail so that everybody can see the rationale of what we are doing mathematically, and why we are doing it.