# Approximation Theory (MATH582) Course Detail

Course Name Course Code Season Lecture Hours Application Hours Lab Hours Credit ECTS
Approximation Theory MATH582 Area Elective 3 0 0 3 5
Pre-requisite Course(s)
MATH 136 Mathematical Analysis II or MATH 152 Calculus II or MATH 158 Extended Calculus II or Consent of the instructor
Course Language English Elective Courses Ph.D. Face To Face Lecture, Question and Answer, Problem Solving. Prof. Dr. Sofiya Ostrovska This graduate level course aims to provide math students with the fundamental knowledge of constructive theory of functions. The course includes such topics as uniform approximation by polynomials and trigonometric polynomials, approximation by positive linear operators and by general linear systems. The course provides theoretical background for many problems of numerical analysis, applied mathematics, and engineering. The students who succeeded in this course; understand the concepts of the uniform convergence and uniform approximation, construct approximating sequences of operators, calculate moments and central moments of positive linear operators, in particular Bernstein and Bernstein-type operators, apply the fundamental inequalities of approximation theory, analyze the connection between the structural properties of a function and the possible rate of approximation by (trigonometric) polynomials. Uniform convergence, uniform approximation, Weierstrass approximation theorems, best approximation, Chebyshev polynomials, modulus of continuity, rate of approximation, Jackson?s theorems, positive linear operators, Korovkin?s theorem, Müntz theorems.

### Weekly Subjects and Releated Preparation Studies

Week Subjects Preparation
1 Introduction: uniform convergence of sequences and series. Properties of uniformly convergent sequences. Tests for uniform convergence. [2], Ch. 1, Sec. 1,2
2 Uniform approximation by polynomials and trigonometric polynomials. Weierstrass theorems. [1], Ch. 1, Sec. 1-3.
3 The equivalence of two Weierstrass theorems. Approximation by interpolation polynomials . [1], Ch. 2, Sec. 1-3
4 Polynomials of best approximation. Existence theorems. [2], Ch. 3, Sec. 4
5 Chebyshev alternation property. Chebyshev systems. The Haar condition. [1], Ch. 2, Sec. 4-6 [2], Ch. Sec. 4
6 Uniqueness theorems of the polynomial of best approximation. [2], Ch. 3, Sec.5
7 Polynomials of least deviation: Chebyshev polynomials, their properties. [1], Ch. 2, Sec. 7
8 Inequalities of Bernstein and Markov for the derivatives. [1], Ch. 3, Sec. 2,3
9 Modulus of continuity and classes of functions. Midterm I. [1], Ch. 3, Sec. 5,7
10 Direct Jackson's theorems. [1], Ch. 4, Sec. 1,2
11 Inverse Jackson’s theorems. [1], Ch. 4, Sec. 4 [2], Ch. 6, Sec. 3
12 Approximation by positive linear operators. Korovkin’s theorem. Midterm II. [2], Ch. 3, Sec. 3
13 Central moments. Rate of approximation by positive linear operators. [1], Ch. 3, Sec. 6
14 Müntz theorems on the completeness of power systems. [2], Ch. 6, Sec. 2
15 Review.
16 Final exam.

### Sources

Course Book 1. 1. G. G. Lorentz, “Approximation of functions,” Chelsea, NY, 1986. 2. 2. E. W. Cheney, “Introduction to approximation theory”, Chelsea, NY, 1966 3. 3. Ph. J. Davis, “Interpolation and approximation”, Blaisdell NY, 1963. 4. 4. R. DeVore, G. G. Lorentz, “Constructive approximation”, Springer, 1986.

### Evaluation System

Attendance/Participation - -
Laboratory - -
Application - -
Field Work - -
Special Course Internship - -
Quizzes/Studio Critics - -
Homework Assignments 2 10
Presentation 1 10
Project - -
Report - -
Seminar - -
Midterms Exams/Midterms Jury 2 40
Final Exam/Final Jury 1 40
Toplam 6 100
 Percentage of Semester Work 60 40 100

### Course Category

Core Courses X

### The Relation Between Course Learning Competencies and Program Qualifications

# Program Qualifications / Competencies Level of Contribution
1 2 3 4 5
1 Ability to carry out advanced research activities, both individual and as a member of a team
2 Ability to evaluate research topics and comment with scientific reasoning
3 Ability to initiate and create new methodologies, implement them on novel research areas and topics
4 Ability to produce experimental and/or analytical data in systematic manner, discuss and evaluate data to lead scintific conclusions
5 Ability to apply scientific philosophy on analysis, modelling and design of engineering systems
6 Ability to synthesis available knowledge on his/her domain to initiate, to carry, complete and present novel research at international level
7 Contribute scientific and technological advancements on engineering domain of his/her interest area
8 Contribute industrial and scientific advancements to improve the society through research activities

Activities Number Duration (Hours) Total Workload
Course Hours (Including Exam Week: 16 x Total Hours)
Laboratory
Application
Special Course Internship
Field Work
Study Hours Out of Class 14 3 42
Presentation/Seminar Prepration 1 7 7
Project
Report
Homework Assignments 2 2 4
Quizzes/Studio Critics
Prepration of Midterm Exams/Midterm Jury 2 7 14
Prepration of Final Exams/Final Jury 1 10 10