Bernstein Polynomials (MATH555) Course Detail

Course Name	Course Code	Season	Lecture Hours	Application Hours	Lab Hours	Credit	ECTS
Bernstein Polynomials	MATH555	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 is designed to provide math students with the knowledge of basic facts about the Bernstein polynomials and their role in analysis and approximation theory, as well as demonstrate their applications and generalizations. For this purpose, the course includes topics on positive linear operators, Kantorovich polynomials, and the De Casteljau algorithm, which are closely related to the Bernstein polynomials.
Course Learning Outcomes	The students who succeeded in this course; understand the notions of uniform continuity and uniform approximation; construct the Bernstein, Chlodovsky and Kantorovich polynomials of functions given on different intervals; apply Korovkin’s theorem to establish the approximation by a sequence of positive linear operators; use the Bernstein polynomials in applied problems; understand shape-preserving and degree-reducing properties of the Bernstein polynomials.
Course Content	Uniform continuity, uniform convergence, Bernstein polynomials, Weierstrass approximation theorem, positive linear operators, Popoviciu theorem, Voronovskaya theorem, simultaneous approximation, shape-preserving properties, De Casteljau algorithm, complex Bernstein polynomials, Kantorovich polynomials.

Weekly Subjects and Releated Preparation Studies

Week	Subjects	Preparation
1	Uniform continuity, Cantor’s theorem. Uniform convergence of sequences and series.	[2], Ch. 1, Sec. 1.5-1.6
2	Properties of uniformly convergent sequences. Tests for uniform convergence.	Davis, Ch. 1, Sec. 1.6-1.7
3	Bernstein polynomials, their definition and elementary properties. Weierstrass approximation theorem.	[1], Ch. 1, Sec. 1.1, [2], Ch. VI, Sec. 6.1,6.2
4	Positive linear operators, Korovkin’s theorem. Modulus of continuity and its properties.	[2], Ch. 6, Sec.6.6
5	Moments and central moments. Popoviciu theorem.	[2], Ch. 1, Sec. 1.6
6	Voronovskaya theorem and modified Bernstein polynomials.	[2], Ch. 6, Sec. 6.3, [1], Ch. 1, Sec. 1.6
7	Forward differences representation of the Bernstein polynomials and their derivatives.	[1], Ch. 1, Sec. 1.4
8	Simultaneous approximation of a function and its derivatives by the Bernstein polynomials.	[2], Ch. 6, Sec. 6.3, [1], Ch. 1, Sec. 1.8
9	Shape-preserving properties of the Bernstein polynomials.	[1], Ch. 1, Sec. 1.7
10	De Catseljau algorithm for the Bernstein polynomials.	[3], Sec.2
11	Bernstein polynomials on an unbounded interval. Chlodovsky’s theorems.	[1], Ch. 2, Sec. 2.3
12	Complex Bernstein polynomials.	[1], Ch. 4, Sec. 4.1
13	Kantorovich polynomials, their properties.	[1], Ch.2, Sec. 2.1
14	Approximation of continuous and integrable functions by Kantorovich polynomials.	[1], Ch.2, Sec. 2.2
15	Review
16	Final exam

Sources

Course Book	1. [1] G. G. Lorentz, Bernstein polynomials, Chelsea, NY, 1986.
	2. [2] Ph. J. Davis, Interpolation and Approximation, Dover, 1976.
Other Sources	3. W. Boehm, A. Müller, On de Casteljau's algorithm,
	4. 2. R.A.Devore, G.G.Lorentz, Constructive Approximation, Springer,
	5. E. W. Cheney, “Introduction to approximation theory”, Chelsea, NY, 1966

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	Alanında, bağımsız olarak, bir problem kurgulayabilir, çözüm yöntemi geliştirerek problemi çözebilir ve sonuçları değerlendirebilir				X
2	Matematiğin temel alanlarında ve kendi uzmanlığı olarak seçtiği alanda gerekli alt yapıyı oluşturur.					X
3	Matematik literatürünü ve özel olarak kendi araştırma konusu ile ilgili ulusal ve uluslararası güncel yayınları takip edebilir ve bunlardan kendi araştırma konusu ile ilgili olanları çalışmalarında kullanabilir				X
4	Bilimsel etik değerleri ve kuralları dikkate alır ve mesleki ve toplumsal yaşamda kullanabilir					X
5	Kendi çalışmalarının sonuçlarını veya belli bir konudaki güncel çalışmaları ve bulguları, çeşitli bilimsel toplantılarda topluluk önünde Türkçe ve İngilizce olarak sunabilir ve tartışmalara katılabilir.				X
6	Gerek bireysel, gerek bir çalışma grubunun üyesi olarak çalışabilme becerisini geliştirir				X
7	Yaratıcı ve eleştirel düşünme, problem çözme, özgün bir çalışma üretme becerisini geliştirir. Bilimsel gelişmeleri takip eder, özümsediği bilgilerin analiz, sentez ve değerlendirmesini yapabilir.					X
8	Kazandığı bilgi, beceri ve yetkinlikleri yaşam boyu geliştirmeye açık olur.				X
9	Alanında özümsediği bilgiyi ve problem çözme yeteneğini disiplinler arası çalışmalarda uygulayabilir; karşılaşılan problemleri matematiksel modellerle ifade ederek, matematiksel bakış açısı ile farklı çözüm yöntemleri önerir.				X
10	Matematik temelli yazılımları, bilişim ve iletişim teknolojilerini bilimsel amaçlı kullanabilir.					X

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	5	5
Project
Report
Homework Assignments	2	3	6
Quizzes/Studio Critics
Prepration of Midterm Exams/Midterm Jury	2	7	14
Prepration of Final Exams/Final Jury	1	10	10
Total Workload			77