Bernstein Polynomials (MATH555) Course Detail

Course Name Course Code Season Lecture Hours Application Hours Lab Hours Credit ECTS
Bernstein Polynomials MATH555 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
Course Type N/A
Course Level Natural & Applied Sciences Master's Degree
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


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

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.

ECTS/Workload Table

Activities Number Duration (Hours) Total Workload
Course Hours (Including Exam Week: 16 x Total Hours)
Special Course Internship
Field Work
Study Hours Out of Class 14 3 42
Presentation/Seminar Prepration 1 5 5
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