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 | Natural & Applied Sciences Master's Degree |
| Mode of Delivery | Face To Face |
| Learning and Teaching Strategies | Lecture, Question and Answer, Problem Solving. |
| Course Lecturer(s) |
|
| 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;
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| 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 | |
|---|---|
| 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. | X | ||||
| 2 | Has the ability to obtain, to evaluate, to interpret and to apply information by doing scientific research. | X | ||||
| 3 | Can apply gained knowledge and problem solving abilities in inter-disciplinary research. | X | ||||
| 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. | X | ||||
| 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. | X | ||||
| 6 | Can develop strategies, implement plans and principles on the area of study and can evaluate obtained results within the framework. | X | ||||
| 7 | Can develop and extend the knowledge in the area and to use them with scientific, social and ethical responsibility. | X | ||||
| 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. | X | ||||
| 9 | Has proficiency in English language and has the ability to communicate with colleagues and to follow the innovations in mathematics and related fields. | X | ||||
| 10 | Has software and hardware knowledge in the area of expertise, and has proficient information and communication technology knowledge. | X | ||||
| 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. | 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 | ||