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%. A linear scale for grades will be used with no curving. Grading scale for UCD 0- 21- 28- 35- 41- 48- 55- 61- 68- 75- 81- 91- % 20% 27% 34% 40% 47% 54% 60% 67% 74% 80% 90% 100% UCD F D- D D+ C- C C+ B- B B+ A- A TEXTBOOKS: Format:Textbook Paperback, 1st ed., 352pp. Approximation Theory ISBN: 0521295149 Publisher: Cambridge and Methods, University Press. Pub. Date: October 1981 by M. J. J. D. Powell Price at: BN Format: Hardcover, 2nd ed., 188pp. 1986 Approximation of ISBN: 0828403228 Publisher: American Functions, Mathematical Society Pub. Date: December by George G. Lorentz 1997 Price at: BN An Introduction to the Format: Paperback, 160pp. ISBN: 0486640698 Approximation of Publisher: Dover Publications, Incorporated Functions, Pub. Date: November 1987 Price at: Amazon by T. J. Rivlin 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: Ch. 1. The approximation problem and existence of best approximations; Jan 20 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. LINKS: 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 Similar courses at other universities: University of Wisconsin-Madison University of Leicester, UK University of Auckland, NZ REQUIRED INFORMATION FROM CLAS .