COURSE SYLLABUS

Course Number: MATH 7050
Course Title: APPROXIMATION THEORY II
Credit Hours: 3
Prerequisites: MATH7040.
Corequisite: 

Objectives: 
This sequence is intended to serve as an introduction at the graduate level into the subject of approximation, in which techniques from several different areas of mathematics are used to study the feasibility, efficacy, and efficiency of various methods of approximation. We intend to present the theoretical background which justifies and explains many of the techniques and algorithms in common use and to prepare our students to make future advances and improvements in this area of mathematics. The sequence should also be of interest to students from other
disciplines with a high mathematical content, owing to the universality of problems of approximation. 

Course Content:
1. LEAST SQUARES APPROXIMATION AND RELATED TOPICS (15 hours)
Orthonormal systems, orthogonal polynomials, orthogonal expansions, properties of orthogonal expansions when the uniform norm is used, normal equations, Gram-Schmidt procedure for orthogonalization, Gram matrices and determinants, the theorems of D. Jackson.
2. RATIONAL APPROXIMATION (8 hours)
Existence of best rational approximations the characterization of best rational approximation, uniqueness for generalized rational approximation.
3. SOME ADDITIONAL TOPICS (at the choice of the instructor as time permits; approximately 17 hours allotted). 
Topics would be chosen which lead to research in areas of current interest. 
 
Possible Textbooks: Unfortunately, there are not many current books which would serve well as texts. There are many recent results in the field which are not in textbooks yet. Some classic texts which might serve well are the following:
1. E. W. Cheney, Introduction to Approximation Theory, Second Edition, Chelsea Publishing Company, New York, 1982.
2. Philip J. Davis, Interpolation and Approximation, Blaisdell Publishing Company, New York, 1963.
3. I. P. Natanson, Constructive Function Theory, Frederick Ungar Publishing Company, New York, 1964.
4. A. F. Timan, Theory of approximation of functions of a real variable, Oxford University Press, 1963.
   (paperback reprint edition by Dover, 1993)

Sample Grading and Evaluation Procedures

A graduate student is expected to creatively engage the mathematical material of the course.  The student will be given problems to solve; these problems may include the derivation of proofs to theorems.  These solutions may be presented in class on the blackboard or be written up to be handed in to the instructor. Extended projects may also be assigned.  

Grade Calculation
Presentation of solutions to problems/theorems, homework:				40%
Midterm exam or midterm project:							25%
Final Exam or culminating project:							35%

There may be variations in these procedures depending on the individual instructors and the nature of the specific material.

Sample Statement Re: Accommodations
	Students who need accommodations are asked to arrange a meeting during office hours the first week of classes, or as soon as possible if accommodations are needed immediately.  If you have a conflict with my office hours, an alternate time can, be arranged.  To set up this meeting, please contact me by E-mail.  Bring a Copy of your Accommodation Memo and an Instructor Verification Form to the meeting.  If you do not have an Accommodation Memo but need accommodations, make an appointment with The Program for Students with Disabilities, 1244 Haley Center, 844-2096 (V/TT).  (Note: Instructor office room, office hours and email address will be made available on the course syllabus and on the first day of class.)

JUSTIFICATION FOR GRADUATE CREDIT

	This course is part of a modified semester version of a 600-level sequence under the quarter system.  Under the quarter system it was specifically designed as a graduate course.  It was approved as part of our graduate program and has been a traditional part of our graduate program offerings.  Outside of modernization, the standard of the course remains at the same graduate level that the department has maintained in the past. The course demands considerable mathematics background and a degree of mathematical maturity traditionally found at the graduate level.  The 7000-level course will inculcate the same analytical skills and depth of understanding previously demanded by the comparable 600-level quarter course.  In order to successfully complete the course the student will have to demonstrate an ability to creatively examine and apply the mathematics presented in the course.