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)
N/A

Course Language	English
Course Type	Elective Courses
Course Level	Ph.D.
Mode of Delivery	Face To Face
Learning and Teaching Strategies	Lecture, Question and Answer, Problem Solving.
Course Coordinator
Course Lecturer(s)	Prof. Dr. Sofiya Ostrovska
Course Assistants
Course Objectives	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.
Course Learning Outcomes	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.
Course Content	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
Other Sources	3. 3. Ph. J. Davis, “Interpolation and approximation”, Blaisdell NY, 1963.
	4. 4. R. DeVore, G. G. Lorentz, “Constructive approximation”, Springer, 1986.

Evaluation System

Requirements	Number	Percentage of Grade
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
Percentage of Final Work	40
Total	100

Course Category

Core Courses	X
Major Area Courses
Supportive Courses
Media and Managment Skills Courses
Transferable Skill Courses

The Relation Between Course Learning Competencies and Program Qualifications

#	Program Qualifications / Competencies	Level of Contribution
#	Program Qualifications / Competencies	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

ECTS/Workload Table

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
Total Workload			77