TITLE: Local Approximations Based on Orthogonal Differential Operators PRESENTER: Aleksandar Ignjatovic, http://www.cse.unsw.edu.au/~ignjat, ignjat@cse.unsw.edu.au AFFILIATION:School of Computer Science and Engineering, UNSW, http://www.cse.unsw.edu.au DATE: Friday 24th August 2007 TIME: 12:00:00 PLACE: CSE Seminar Room, Level 1, K17 ABSTRACT: This is a replay of the talk I gave at the Analysis Seminar (School of Mathematics at UNSW), but was asked for a replay by two colleagues. I will present some generalizations of the Neumann expansion of analytic functions (as a series of Bessel functions), called the chromatic expansions. Like truncations of a Taylor expansion, truncations of a chromatic expansion are local approximations; they converge uniformly for important classes of analytic functions.The coefficients of a chromatic expansion of an analytic function f(t) are of the form K_n[f](0), where K_n are linear differential operators, orthogonal with respect to a suitably defined scalar product. A family of such orthogonal operators K_n can be described using a three-term recurrence formula, akin to the recurrence formulas for families of orthogonal polynomials. We relate the class of analytic functions that can be represented by their chromatic expansions to the asymptotic growth rate of the recursion coefficients involved in such a corresponding recurrence. Unlike the derivatives of high order, the values of K_n[f](t) can be approximated in a numerically robust way using the values of discrete samples of f(t). This could make the chromatic approximations useful in practical applications, such as signal processing.This talk is a summary of my paper "Local Approximations Based onOrthogonal Differential Operators" forthcoming in the Journal ofFourier Analysis and Applications; the preprint is available athttp://www.cse.unsw.edu.au/~ignjat/diff/LocalApprox.pdf. BIOGRAPHY OF SPEAKER: Dr. Ignjatovic is Senior Lecturer at the School of Computer Science and Engineering (CSE), University of New South Wales (UNSW). He joined CSE in 2002 where he is teaching algorithms and data structures. His research interests include applications of mathematical logic to computational complexity theory, sampling theory and signal processing, algorithms for embedded systems design as well as educational use of puzzles for teaching serious problem solving techniques. After graduating at Berkeley, Dr. Ignjatovic got a tenure track positionas an Assistant Professor at the Carnegie Mellon University, where hetaught for 5 years at the Department of Philosophy and the CMU's Program for Pure and Applied Logic. He left CMU to start a company, and his startup "Kromos Technology" employed several of his former CMU students. The company's CEO was Raj Parekh, former CTO of Sun Microsystems and among our investors and Board members were former President and COO of AMD Atiq Raza, the former CEO of Fiberlane, Cerent and Siara Raj Singh,as well as Redwood Venture Partners. The company was acquired by Comstellar Technologies and it is now a part of "Kromos Telecommunications". Seminar information is also available at http://www.cse.unsw.edu.au/db/ai/seminars/list/index.html Host: Seminar Convenor:
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