# Approximation Theory (MATH365) Course Detail

Course Name Course Code Season Lecture Hours Application Hours Lab Hours Credit ECTS
Approximation Theory MATH365 3 0 0 3 6
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 N/A Bachelor’s Degree (First Cycle) Face To Face Lecture, Question and Answer, Problem Solving. Prof. Dr. Sofiya Ostrovska The objective of the course is to study basic notions of Approximation Theory. Approximation Theory not only provides theoretical foundations for Applied Mathematics, Numerical Analysis, and Scientific Computing, but also gives methods to solve practical problems of computation. The course is for students of mathematical and engineering departments interested in analysis and its applications to numerical computations. The students who succeeded in this course; At the end of the course the students are expected to: 1) Understand the notions of interpolation as well as of uniform and least-square approximation. 2) Be able to analyze inconsistent linear system and find their Chebyshev solutions. 3) Know the Weierstrass approximation theorem and Bernstein polynomials. 4) Understand the notion of convexity and knowledge of the Caratheodory theorem. 5) Knowledge of various orthogonal systems of functions and orthogonal expansions. Preliminaries, Convexity, Chebychev Solution of Inconsistent Linear Systems, Interpolation, Approximation of Functions by Polynomials, Least-Squares Approximation.

### Weekly Subjects and Releated Preparation Studies

Week Subjects Preparation
1 Metric spaces. Normed linear spaces. The space C[a,b]. Inner-product spaces. The Gram-Schmidt process. [1] Ch. I, pp. 3-16
2 Convex sets. Caratheodory’s theorem. [1] Ch. 1, pp. 16-20
3 Convex functions: local and absolute extrema, continuity. Existence and unicity of the best approximation. [1] Ch. I, pp. 20 - 27
4 A minimax solution of a linear system. Inconsistent systems of linear equations with one unknown, their graphical solution. [1] Ch. 2, pp. 28 – 33
5 Characterization of Chebychev solutions. The ascent and descent algorithms. [1] Ch. 2, pp. 34-37, pp. 45-56
6 Lagrange interpolation polynomial. Error formula. Hermite interpolation. [1] Ch. 3, pp. 57-60, 62-65
7 Review and Midterm I
8 The Weierstrass approximation theorem. [1] Ch. 3, pp. 61 - 67
9 Monotone operators. Korovkin’s theorem. [1] Ch. 3, pp. 65-71
10 Bernstein polynomials. [3] Ch. VI, pp. 108-111
11 Polynomials of the best approximation. Alternation theorem. Orthogonal systems of polynomials, their properties. [1] Ch. 3, pp. 72-77, Ch. 4, pp. 101 - 105
12 Review and Midterm II
13 Uniform and least-squares convergence, Christoffel-Darboux identity, Bessel Inequality [1] Ch. 4, pp. 115 - 119
14 Convergence of Fourier series, Fejer’s theorem. [1] Ch. 4, pp. 120 - 125
15 Review
16 Final exam.

### Sources

Course Book 1. [1] E.W. Cheney. Introduction to Approximation Theory. Chelsea Publ. 2. [2] G. G. Lorentz, Approximation of Functions, AMS Chelsea publishing, 1986. 3. [3] P. J. Davis, Interpolation and Approximation, Dover Publications, 1975

### Evaluation System

Attendance/Participation - -
Laboratory - -
Application - -
Field Work - -
Special Course Internship - -
Quizzes/Studio Critics - -
Homework Assignments 4 20
Presentation - -
Project - -
Report - -
Seminar - -
Midterms Exams/Midterms Jury 2 40
Final Exam/Final Jury 1 40
Toplam 7 100
 Percentage of Semester Work 60 40 100

### Course Category

Core Courses X

### 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 Can apply gained knowledge and problem solving abilities in inter-disciplinary research X
3 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
4 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
5 Can develop strategies, implement plans and principles on the area of study and can evaluate obtained results within the framework X
6 Can develop and extend the knowledge in the area and to use them with scientific, social and ethical responsibility X
7 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
8 To have an oral and written communication ability in at least one of the common foreign languages ("European Language Portfolio Global Scale", Level B2) X
9 Has software and hardware knowledge in the area of expertise, and has proficient information and communication technology knowledge X
10 Follows scientific, cultural, and ethical criteria in collecting, interpreting and announcing data in the research area and has the ability to teach. 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

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 16 3 48
Presentation/Seminar Prepration
Project
Report
Homework Assignments 4 10 40
Quizzes/Studio Critics
Prepration of Midterm Exams/Midterm Jury 2 12 24
Prepration of Final Exams/Final Jury 1 18 18