MATH 5667-001, C SC 5667-001: Introduction to Approximation Theory, Fall 2006, UCDHSC

PREREQUISITE: MATH 4320: ADVANCED CALCULUS II.
MATH 5070 APPLIED ANALYSIS is not formally required but is recommended.

CLASS DAY/TIME/LOCATION: Tuesday, Thursday 2:30-3:45, CU-Dravo 641.

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

MATH DEPT ASSOCIATE CHAIR: Lynn Bennethum CU-Bldg., Rm. 638 Ph.303-556-4810.

OFFICIAL COURSE DESCRIPTION: Introduction to Approximation Theory. Normed linear spaces, convexity, existence and uniqueness of best approximations. Tchebychev approximation by polynomials and other related families. Least squares approximation and splines.

COURSE ADVERTISEMENT: For analysis lovers and admirers of Smirnoff and Tchebycheff. Tired of textbooks with color pictures, and want to read a real text with more formulas than words published by Dover more than twenty years ago that still costs $8.95? Looking for exciting computer projects comparing apples and oranges? Then consider using this rare opportunity to learn the Approximation Theory, one of the cornerstones of numerical mathematics. The class, MATH/C SC 5667-001 : Introduction to Approximation Theory, has been only offered twice before at CU-Denver in 1998, see http://math.cudenver.edu/~aknyazev/teaching/98/5667/ and in 2002, see http://math.cudenver.edu/~aknyazev/teaching/02/5667/.

SUBJECT: A survey of classical techniques in Approximation Theory.

GRADING: 4 Homework assignments - 10% each, Midterm Project - 20%, Final Project - 20%, Final Test (in class) - 20%.

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 $48
	Approximation of Functions, by George G. Lorentz		Format: Hardcover, 2nd ed., 184pp. ISBN: 0828403228 Publisher: American Mathematical Society Pub. Date: December 1997 Price at: BN $23
	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 $9

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.

• 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 Dundee, UK
• University of Auckland, NZ

Drops and incompletes: You have until October 30th to drop the course with only the instructor's (but not a Dean's) signature. The incomplete policy of the department and college is strictly enforced. Incompletes are given only in situations in which a student has: (1) Successfully completed 75 percent of the course (i.e is passing the course)(2) Special circumstances (verification may be required) that preclude the student from attending class and completing graded assignments, and (3) Made arrangements to complete missing assignments with the original instructor. A CLAS Course Completion agreement is required.

Fall 2006 Registration and Academic Deadlines, and Important Policies

• CLAS students must always have an accurate mailing and e-mail address: Go to http:/www.cudenver.edu/registrar to update and/or change.
• Students must complete and submit a drop/add form to make any schedule changes. Students are not automatically dropped from a class if they stop attending or do not make tuition payments. The student is ultimately responsible for verifying their schedule prior to officially published drop dates.
• Late adds will be approved only when circumstances surrounding the late add are beyond the student's control and can be documented. This will require a petition and documentation from the student.
• Late drops will be approved only when circumstances surrounding the late drop are beyond the student's control and can be documented. This will require a petition and documentation from the student.
• Students who wish to graduate in December of 2006 MUST meet with their academic advisor to obtain a graduation application. The application must be completed and submitted by September 6, 2006.
• Students are responsible for completing financial arrangements with financial aid, family, scholarships, etc. to pay their tuition. Students will be responsible for all tuition and fees for courses they do not officially drop using proper drop/add procedures and forms.
• August 24, 2006 (midnight) Last day to be added to the wait-list for a closed course.
• August 24 - September 6, 2006 Students are responsible for verifying an accurate fall 2006 registration via SMART. Students are NOT notified of their wait-list status by the University. All students must check their schedules prior to September 6, 2006 for accuracy.
• August 31, 2006 (midnight) Last day to add courses via the web SMART system.
• September 6, 2006 (5:00 pm) Last day to add structured courses without a written petition for a late add. This is an absolute deadline. This deadline does not apply to independent study, internships, and late-starting modular courses.
• September 6, 2006 (5:00 pm) Last day to drop a fall 2006 course for tuition refund and no transcript notation. This is an absolute deadline.
• September 6, 2006 (5:00 pm) Last day for undergraduates and graduates to apply for December 2006 graduation. This is an absolute deadline.
• September 6, 2006 (5:00 pm) Last day to request pass/fail or no credit option. This is an absolute deadline.
• October 30, 2006 (5:00 pm) Last day for NON-CLAS students to drop a fall 2006 course or withdraw from all classes without a petition to their home college and receiving their Dean's approval.
• November 10, 2006 (5:00 pm) Last day for CLAS students to drop a fall 2006 course or withdraw from all classes. Treated as an absolute deadline. Dean's approval required.
• See Academic Calendar for details on registration/payment deadlines: http://www.cudenver.edu/registrar

Other resources for this course:

The Center for Learning Assistance
The Center for Learning Assistance is where students go to get help or insight with class assignments, course-loads, and study skills. The Center also helps with arranging tutoring sessions, which take place in the days or evenings. North Classroom Building (NC) Room 2006 (303) 556-2802 Monday - Thursday 8am-7pm Friday 8am-5pm Program Access for Persons with Disabilities The University of Colorado at Denver is committed to providing reasonable accommodations and access to programs and services to persons with disabilities. Students should contact the Disabilities Services Offices.
Arts building 177; 303-556-3450, TTY 303-556-8484.

Academic Advising Center
This office serves as the first point of contact for students who are pre-business, pre-engineering, or who have not declared a major in CLAS or CAM. In addition, the center provides general information and resource referral to all students.
North Classroom Building (NC) Room 1 Phone # 303-352-3520

Career Center
The Career Center offers a full array of services that prepare students for career success, such as resume help, internship and career counseling and they have a large career library.
Tivoli Student Union Room 260