ECTS - Bernstein Polynomials
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 Lecturer(s) |
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| 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;
|
| 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 | ||||
|---|---|---|---|---|---|---|
| 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 | 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 | ||