Back to Departmental Colloquium: Fall 2004
Departmental Colloquium
Date: Thursday, Oct 21, 2004
Time: 4:15PM
Location: JWB 335
Special Colloquium
High Resolution Algorithms Through Localized Fourier Analysis
Title |
High Resolution Algorithms Through Localized Fourier Analysis |
Abstract |
Although global projections such as truncated Fourier series yield exponentially close approximations for smooth functions, a single discontinuity introduces O(1) spurious oscillations, Gibbs’ Phenomena, and reduces the high order convergence rate to first order. A family of filters with different properties have been developed over the last century to reduce the effects of the Gibbs phenomenon; however, which filter to use for a given application has remained largely heuristic. In the first half of the talk I construct a spatially adaptive filter which is shown to achieve optimal (exponential) accuracy for this class of methods, and as a result definitively resolves the question of filter selection. Second I consider a problem in modern communication and signal processing, the recovery of a bandlimited signal >From its bunched samples. In many emerging applications the uniform sampling required for the classical Shannon sampling theorem is not realizable, and instead bunched sampling is utilized. Here I present an efficient and robust high order algorithm for the “bunched” sampling structure, with the same exponential accuracy as can be achieved with uniform sampling. Portions of these research projects were conducted in collaboration with Eitan Tadmor and Thomas Strohmer. |
Computational Mathematics
Applied Mathematics
Data Science and Machine Learning