# Approximation Theory

Engineering, finance, science, and many areas of mathematics itself make use of quantities that are too complicated, too difficult, and even too abstract to work with directly. A major goal of approximation theory is to discover and analyze simple, easy to work with, concrete quantities that can do a good, efficient job in their place - for example, splines to fit messy curves, wavelets to analyze noisy signals and to compress large images, and radial basis functions to fit scattered data and serve as the ``approximation engine'' of neural networks.

## Graduate Program

The graduate program in approximation theory includes basic courses on splines MATH 657 and on foundations and methods of approximation theory MATH 667, and advanced courses on applied harmonic analysis MATH 658 and wavelets MATH 668, as well as courses on a variety of special topics.

Areas of faculty interest include radial and related basis functions, scattered data surface fitting, rates of approximation, constrained approximation, polynomial inequalities, wavelets, splines, and a variety of other topics and fields. A list of involved faculty and some areas each works in is included below, along with current PhD graduate students.

## Faculty

G. Donald Allen
Transport and diffusion, splines, and applied mathematics

Guy Battle
Wavelets, multiwavelets, renormalization group, and mathematical physics

Itshak Borosh
Combinatorics, number theory, complexity

Charles K. Chui (Emeritus)
Splines, wavelets, applied harmonic analysis, signal and image processing, classical analysis

Tamás Erdélyi
Polynomials, polynomial inequalities, orthogonal polynomials, Chebychev and Descartes systems, and classical analysis

David Larson
Wavelets, frames, operator theory, and linear analysis

Francis J. Narcowich
Radial basis functions, positive definite functions on manifolds, approximation and interpolation on spheres, quadrature, scattered-data surface fitting, neural networks, wavelets, and mathematical physics

Guergana Petrova
Nonlinear approximation, hyperbolic PDEs, conservation laws, numerical quadrature on balls in Rn.

Bojan Popov
Conservation laws, linear transport equations, approximation theory, numerical analysis of PDEs

N. Sivakumar
Splines, radial basis functions, approximation and interpolation on spheres, classical analysis, cardinal interpolation

Joseph D. Ward
Radial basis functions, approximation and interpolation on spheres, quadrature, scattered-data surface fitting, constrained approximation, wavelets

## Graduate Students

Edward Fuselier
Radial basis functions

## Recent Ph.D.'s

Svenja Lowitzsch
Dissertation: Approximation and interpolation employing divergence-free radial basis functions with applications

Quoc Thong Le Gia
Dissertation: Approximation of linear partial differential equations on spheres
Updated: 5/20/2005 (fjn)