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) |
|
| 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 | Is independently able to build a problem in the area of study, solve the problem by developing solution techniques and assess the solutions. | X | ||||
| 2 | Is capable of creating a groundwork in the fundamental branches of mathematics as well as in his/her research area | X | ||||
| 3 | follows the latest national and international literature in Mathematics and in his/her area of research; and uses them in his/her related studies | X | ||||
| 4 | observes and adopts the scientific ethical values in his/her professional and social life | X | ||||
| 5 | presents in Turkish and English in academic/scientific events the results of his/her research or the latest studies and findings on a special topic and participates in discussions | X | ||||
| 6 | Develops skills to work independently or as a member of a team | X | ||||
| 7 | Develops competences in the areas of creative and critical thinking, problem solving and producing original studies. Follows recent scientific studies, is capable of making an analysis, synthesis and assessment of the knowledge acquired | X | ||||
| 8 | Is open to lifelong improvement of his/her acquired knowledge, skills and competences. | X | ||||
| 9 | Is able to apply the acquired knowledge and problem-solving skills to interdisciplinary studies, proposes different solution methods to problems in terms of mathematical models and from a mathematical point of view | X | ||||
| 10 | Uses the mathematical based softwares, informatics and communication technologies for scientific purposes | 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 | ||