Approximation Theory

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

Professor Robert Sharpley
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.
  (Use Acrobat Reader to see markups and comments).