Current
enrollment
PREREQUISITE:
MATH 4320: ADVANCED CALCULUS II.
MATH 5070 APPLIED ANALYSIS is not
formally required but is strongly recommended.
HOURS: Tue 5:30-8 pm, CU-Dravo 641.
INSTRUCTOR:
Prof. Andrew Knyazev
Office: CU (Dravo Bldg) 644. Phone: (303) 556-8102.
Office hours: by appointment.
WWW: http://www-math.cudenver.edu/~aknyazev
ADVERTIZEMENT FOR STUDENTS:
Are you an analysis lover, who cruised through 5070 with a good grade and
is looking for more challenges in analysis?
Perhaps, have heard of
Smirnoff, but not sure about
Tchebycheff?
Tired of those boring textbooks on glassy paper with color pictures and large print,
and want to read instead a real textbook with more formulas than words
published by Dover more than twenty years ago, which still costs $8.95?
Then, perhaps you should consider using this unique opportunity to learn
the Approximation Theory, one of the cornerstones of numerical mathematics.
This class, MATH 5667-001 : Introduction to Approximation Theory,
was only offered ones before at CU-Denver in 1998, see
http://math.cudenver.edu/~aknyazev/teaching/98/5667/.
It is offered again this fall of 2002, see
http://math.cudenver.edu/~aknyazev/teaching/02/5667/
for details. Please register earlier if you can,
so the class would not be in danger of being canceled
because of the low enrollment.
TEXTBOOKS:
-
Approximation Theory and Methods,
M. J. J. D. Powell.
Format:Textbook Paperback, 1st
ed., 352pp.
ISBN: 0521295149
Publisher:
Cambridge University Press.
Pub. Date: October 1981
Price at:
BN $32
-
Approximation of Functions, George G. Lorentz.
Format: Hardcover, 2nd ed., 184pp.
ISBN: 0828403228
Publisher:
American Mathematical Society
Pub. Date: December 1997
Price at:
BN $17
- An Introduction to the Approximation of Functions, T. J. Rivlin.
Format: Paperback, 160pp.
ISBN: 0486640698
Publisher: Dover Publications, Incorporated
Pub. Date: November 1987
Price at:
Amazon $9
SUBJECT:
A survey of classical techniques in Approximation Theory.
CONTENTS:
The class will follow the outline below, touching on each major topic
in a depth that will be determined by the pace of the class.
Approximation Theory and Methods by Powell
will be used as the main textbook for the class, with the
following chapters covered:
-
Ch. 1. The approximation problem and existence of best approximations;
-
Ch. 2. The uniqueness of best approximations;
-
Ch. 3. Approximation operators and some approximating functions;
-
Ch. 4. Polynomial interpolation;
-
Ch. 5. Divided differences;
-
Ch. 6. The uniform convergence of polynomial approximations;
-
Ch. 7. The theory of minimax approximation;
-
Ch. 11. Least squares approximation;
-
Ch. 12. Properties of orthogonal polynomials;
-
Ch. 13. Approximation of periodic functions;
-
Ch. 16. The order of convergence of polynomial approximations;
-
Ch. 17. The uniform boundedness theorem;
-
Ch. 18. Interpolation by piecewise polynomials;
Approximation of Functions by Lorentz is more advanced.
A brief review of the following chapters will be presented if time
allows:
-
Ch. 1. Possibility of Approximation:
1. Basic notions; 2. Linear operators; 3. Approximation theorems;
-
Ch. 2. Polynomials of Best Approximation;
1. Existence of polynomials of best approximation; 2. Characterization of polynomials of best approximation; 3. Applications of convexity; 4. Chebyshev systems; 5. Uniqueness of polynomials of best approximation; 6. Chebyshev's theorem; 7. Chebyshev polynomials;
-
Ch. 3. Properties of Polynomials and Moduli of Continuity:
1. Interpolation; 2. Inequalities of Bernstein; 3. The inequality of Markov;
Finally, the book An Introduction to the Approximation of Functions by
Rivlin is recommended for independent reading, in particular:
-
Ch. 1. Uniform Approximation.
-
Ch. 2. Least-Squares Approximation.
-
Ch. 4. Polynomial and Spline Interpolation.
GRADING:
Midterm Test - 25%,
Final Project - 25%,
Homework assignments - 10% each.
LINKS:
Similar courses at other universities: