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 on
Orthogonal Differential Operators" forthcoming in the Journal of
Fourier Analysis and Applications; the preprint is available at
http://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 position
as an Assistant Professor at the Carnegie Mellon University, where he
taught 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:

Van Hai Ho