MATH 7667-001, Introduction to Approximation Theory

Department of Mathematical & Statistical Sciences

College of Liberal Arts and Sciences

University of Colorado Denver

Class Webpages: http://math.ucdenver.edu/~aknyazev/teaching/11/7667/ and CUonline Blackboard

PREREQUISITES: MATH 5070 and MATH 5718.

CLASS DAY/TIME/LOCATION: Tuesday, Thursday 12:30-1:45, CU Denver 626.

INSTRUCTOR: Prof. Andrew Knyazev http://math.ucdenver.edu/~aknyazev
Office: CU Denver 644. Phone: 303-556-8102. Fax: 303-556-8550 (Attn: Knyazev).
Office hours: by appointment.

OFFICIAL COURSE DESCRIPTION: Linear normed and Banach spaces, convexity, existence and uniqueness of best approximations, least square approximation and orthogonal polynomials, Chebyshev approximation by polynomials and other related families, splines.

SUBJECT: A survey of classical techniques in Approximation Theory.

GRADING: Midterm Project - 25%, Midterm Test (in class) - 25%, Final Project - 25%, Final Test (in class) - 25%.

TEXTBOOKS:

	Approximation Theory and Methods, by M. J. J. D. Powell		Format:Textbook Paperback, 1st ed., 352pp. ISBN: 0521295149 Publisher: Cambridge University Press. Pub. Date: October 1981 Price at: BN
	Approximation of Functions, by George G. Lorentz		Format: Hardcover, 2nd ed., 188pp. 1986 ISBN: 0828403228 Publisher: American Mathematical Society Pub. Date: December 1997 Price at: BN
	An Introduction to the Approximation of Functions, by T. J. Rivlin		Format: Paperback, 160pp. ISBN: 0486640698 Publisher: Dover Publications, Incorporated Pub. Date: November 1987 Price at: Amazon

TENTATIVE CONTENTS:
The class is planned to 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 if time allows:

• 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. 11. Least squares approximation;
• Ch. 12. Properties of orthogonal polynomials;
• 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.

• L_p strict convexity follows from Clarkson_inequalities see the proof at A short course on Banach space theory by N. L. Carothers

• Class NOTES by Carl de Boor
• A survey, NONLINEAR APPROXIMATION by Ron DeVore, that has appeared in Acta Numerica; 7; 1998.
• History of Approximation Theory and its MIRROR
• A Short Course on Approximation Theory by Neal Carothers
• Constructive Approximation

• University of Wisconsin-Madison
• University of Leicester, UK
• University of Auckland, NZ