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Approximation Theory
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**
Math 725, Fall 2005
TTh 12:30 - 1:45 p.m., LeConte 310
**

Department of Mathematics

University of South Carolina

**
Office Hours: **

TTh 2:00 - 3:30 p.m. (preliminary), LeConte 313D

**
Pre- or Co-requiste:**

Real Analysis
(Math 703)

**
Course Description
**

Loosely speaking, Approximation Theory is the study of how
general functions may be approximated or decomposed into
more simple building blocks, such as polynomials, splines,
wavelets, or other special functions. The primary focus is
analyzing how properties, such as smoothness or variation, of the
function govern the rates of convergence of the approximating
classes of functions.

This course is open to entering
graduate students in mathematics with the explicit purpose of
providing a foundation to mainstream classical analysis, as well as
brief exposure to current areas of intensive mathematical activity,
including such areas of computational mathematics as (i) large scale data
analysis for geopotential and Graphical Information Systems, (ii) image
and video processing, (iii) Fourier spectral analysis, and
(iv) the cross fertilization of constructive approximation and stochastic
analysis, now becoming known as Mathematical Learning Theory.

The course has been designed to emphasize the theory of univariate
approximation theory which serves either directly as the basis for
much of numerical analysis or at the very least enters in a critical
way in its development.
Math 725 also covers the necessary background for study in modern
topics in pure and applied mathematics, which include Fourier analysis,
wavelets and multiresolution analysis, and nonlinear approximation,
as well as in closely related areas of numerical analysis and numerical
partial differential equations.

**
Main Course Topics**

Existence, uniqueness and characterization of best approximants in
the uniform, least squares, and other common 'metrics'; elementary
approximants include algebraic (Chebyshev) and trigonometric polynomials,
splines, and rational functions. Approximation Theory is introduced by a
proof of the Weierstrass approximation theorem in order to distinguish
constructive from non-constructive methods, and to lead directly to the
discussion of guaranteed rates of convergence when the smoothness of the
target function (or class of functions) is specified, for example, in
Sobolev or Lipschitz spaces. This study includes (a) the classical Jackson
and Bernstein theorems, (b) interpolation by (algebriac and
trigonometric) polynomials and splines, (c) Chebyshev systems, alternations,
and characterization of generalized polynomials of best approximation in
the uniform norm, and (d) B-splines as preliminary wavelet-like multiresolution
basis.

**
Grading
**

Two Tests (roughly Sept 22, Nov 10) each counting 25%,
Homework at 20%, and a Final Examination (2-5 pm on Dec. 5)
at 30%.

**
Lectures:**

Link to Weekly Outline

**Basic References:**

- G.G. Lorentz, "Approximation of Functions" (2-nd ed), Chelsea Publ.,
New York, 1986.

[ISBN: 0-8284-0322-8] - T.J. Rivlin, "An Introduction to the Approximation of Functions",
Dover, New York, 1969.

[ISBN: 0-486-64069-8]

**Addition Reading:**

- R.A. DeVore and G.G. Lorentz, "Constructive Approximation,"
Grundlehren der Mathematischen Wissenschaften
**303**, Springer-Verlag, Berlin, 1993.

**
Supplementary Notes:**

- C. Bennett and R. Sharpley, "Interpolation of Operators", Academic Press, 1988. Extracted Notes on rearrangement-invariance, Hardy inequalities, Lorentz spaces.
- Notes: Bernstein's negative result on uniform Lagrange interpolation
- R. DeVore and V. Popov,
Interpolation of Besov Spaces,
TAMS, 1988.

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