### Chromatic Derivatives and Approximations Speaker

- Series
- Analysis Seminar
- Time
- Wednesday, November 9, 2011 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Aleks Ignjatovic – University of New South Wales

Chromatic derivatives are special, numerically robust linear
differential operators which provide a unification framework for a
broad class of orthogonal polynomials with a broad class of special
functions. They are used to define chromatic expansions which
generalize the Neumann series of Bessel functions. Such expansions are
motivated by signal processing; they grew out of a design of a switch
mode power amplifier.
Chromatic expansions provide local signal representation complementary
to the global signal representation given by the Shannon sampling
expansion. Unlike the Taylor expansion which they are intended to
replace, they share all the properties of the Shannon expansion which
are crucial for signal processing. Besides being a promising new tool
for signal processing, chromatic derivatives and expansions have
intriguing mathematical properties connecting in a novel way
orthogonal polynomials with some familiar concepts and theorems of
harmonic analysis. For example, they introduce novel spaces of almost
periodic functions which naturally correspond to a broad class of
families of orthogonal polynomials containing most classical
families. We also present a conjecture which generalizes the Paley
Wiener Theorem and which relates the growth rate of entire functions
with the asymptotic behavior of the recursion coefficients of a
corresponding family of orthogonal polynomials.