# Approximation Theory (MATH582) Course Detail

Course Name Course Code Season Lecture Hours Application Hours Lab Hours Credit ECTS
Approximation Theory MATH582 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 N/A Natural & Applied Sciences Master's Degree 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 Has the ability to apply scientific knowledge gained in the undergraduate education and to expand and extend knowledge in the same or in a different area.
2 Has the ability to obtain, to evaluate, to interpret and to apply information by doing scientific research.
3 Can apply gained knowledge and problem solving abilities in inter-disciplinary research.
4 Has the ability to work independently within research area, to state the problem, to develop solution techniques, to solve the problem, to evaluate the obtained results and to apply them when necessary.
5 Takes responsibility individually and as a team member to improve systematic approaches to produce solutions in unexpected complicated situations related to the area of study.
6 Can develop strategies, implement plans and principles on the area of study and can evaluate obtained results within the framework.
7 Can develop and extend the knowledge in the area and to use them with scientific, social and ethical responsibility.
8 Has the ability to follow recent developments within the area of research, to support research with scientific arguments and data, to communicate the information on the area of expertise in a systematically by means of written report and oral/visual presentation.
9 Has proficiency in English language and has the ability to communicate with colleagues and to follow the innovations in mathematics and related fields.
10 Has software and hardware knowledge in the area of expertise, and has proficient information and communication technology knowledge.
11 Has professional ethical consciousness and responsibility which takes into account the universal and social dimensions in the process of data collection, interpretation, implementation and declaration of results in mathematics and its applications.

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