Mathematics Colloquium
Local Analysis Using Spectral Data
Hrushikesh Mhaskar
California Institute of Technology
&
Claremont Graduate University
Friday, May 1, 2015
4:10 – 5 p.m.
Building 53 Room 201
Abstract
It is widely believed that expansion of 2π-periodic functions in Fourier series is not
a good strategy to approximate or analyze the functions in the uniform norm. The
Fourier series does not necessarily converge uniformly, and the Fourier coefficients reveal
no local information about the target function, such as the locations of discontinuities.
Moreover, local “bad” behavior of functions affects the convergence throughout the
period. We will introduce the basic concepts, and describe our research in the direction
of approximation by trigonometric polynomials and wavelet-like representations using
Fourier coefficients to illustrate how easy it is to perform these tasks using spectral
data. We will discuss some applications, and mention some extensions of the theory.
About the speaker: Hrushikesh Mhaskar did his undergraduate studies in Institute of Science, Nagpur,
and received his first M. Sc. in mathematics from the Indian Institute of Technology in Mumbai
in 1976. He received his Ph.D. in mathematics and M. S. in computer science from the Ohio
State University, Columbus, in 1980. He then joined Cal. State L. A., and was promoted to
full professor in 1990. Since retirement in 2012, he has been a visiting associate at California
Institute of Technology, Research Professor at Claremont Graduate University, and occasionally
serves as a consultant for Qualcomm. He has published more than 125 refereed articles in the area
of approximation theory, potential theory, neural networks, wavelet analysis, and data processing.
His book,“Weighted polynomial approximation”, was published in 1997, and the book with Dr. D.
V. Pai, “Fundamentals of Approximation Theory” was published in 2000. He was awarded the
Humboldt Fellowship for research in Germany four times. He was John von Neumann distinguished
professor at Technical University of Munich in 2011. His research has been supported by the National
Science Foundation and the U. S. Army Research Office, the Air Force Office of Scientific Research,
the National Security Agency, and the Research and Development Laboratories.
Refreshments before the talk, 3:30 - 4 p.m.,
in the Math Department Conference Room, 25-208B.