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.
Activities
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, scattereddata
surface fitting, neural networks, wavelets, and mathematical physics
 Guergana Petrova
 Nonlinear approximation, hyperbolic PDEs, conservation laws,
numerical quadrature on balls in R^{n}.
 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, scattereddata surface fitting, constrained
approximation, wavelets
Visiting Faculty

 Edward Fuselier
 Radial basis functions
 Svenja
Lowitzsch
 Dissertation: Approximation and interpolation employing
divergencefree radial basis functions with applications
 Quoc Thong Le Gia
 Dissertation: Approximation of linear partial differential
equations on spheres
Updated: 5/20/2005 (fjn)