A Short Course on Approximation Theory
Please read the
disclaimer and description
before downloading these notes.
This is a set of lecture notes for a short course on Approximation
Theory that I offered to graduate students at
Bowling Green State
University, Bowling Green, Ohio, in the Summer of 1998
(and, in somewhat different form, in 1994 and 1990).
A complementary textbook for the course was T. J. Rivlin's
An Introduction to the Approximation of Functions,
Dover, 1981.
The course was intended as a survey of elementary techniques in
Approximation Theory for novices and
non-experts. Experts in the field seeking
new, original, or research topics should look elsewhere.
This is strictly for beginners!
Prerequisites for a thorough understanding
of the course include:
- A first course in advanced calculus or real analysis
(pointwise and uniform convergence, compactness, etc.).
- A rudimentary knowledge of normed spaces and completeness.
- A course in linear algebra.
There is enough material here for roughly 25 hour-and-a-half
lectures; probably not quite enough for a full semester course.
On the other hand, it is sufficient background to facilitate reading
E. W. Cheney's Introduction to Approximation Theory,
Chelsea, 1982,
or G. G. Lorentz's Approximation of Functions,
Chelsea, 1986, two excellent sources for further study.
The notes are written in Plain TeX (plus AMSFonts) and are
available here in dvi and Postscript format. The printed version is
159 pages.
- Available as a dvi file:
- Macintosh stuffed dvi file, 203K
- UNIX gnu-zipped dvi file, 203K
- Available as a Postscript file:
- UNIX gnu-zipped ps file, 325K
Table of Contents
- Preface
- Preliminaries
- Exercises on Normed Spaces
- Approximation by Algebraic Polynomials
- Exercises on Approximation by Polynomials
- Approximation by Trigonometric Polynomials
- Exercises on Trigonometric Polynomials
- Characterization of Best Approximation
- Exercises on Chebyshev Polynomials
- Simple Application of Chebyshev Polynomials
- Lagrange Interpolation
- Exercises on Interpolation
- Approximation on Finite Sets
- Introduction to Fourier Series
- Exercises on Fourier Series
- Jackson's Theorems
- Orthogonal Polynomials
- Exercises on Orthogonal Polynomials
- Gaussian Quadrature
- The Müntz Theorems
- The Stone-Weierstrass Theorem
- Short List of References
Neal Carothers -
carother@bgnet.bgsu.edu