Fourier Analysis (FSI-SFA)

Academic year 2018/2019
Supervisor: prof. Aleksandre Lomtatidze, DrSc.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech
Aims of the course unit:
The aim of the course is to familiarise students with basic topics and techniques of the Fourier analysis used in other mathematical subjects
Learning outcomes and competences:
Knowledge of basic topics of Fourier Analysis, manely, Fourier series, Fourier and Laplace transformations, and ability to apply this knowledge in practice.
Prerequisites:
Calculus, basic konwledge of linear functional analysis, measure theory.
Course contents:
The course is devoted to basic properties of Fourier Analysis and illustrations of its techniques on examples. In particular, problems on reprezentations of functions, Fourier and Laplace transformations, their properties and applications are studied.
Teaching methods and criteria:
The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.
Assesment methods and criteria linked to learning outcomes:
Participation in the seminars is mandatory.
Course-unit credit is awarded on condition of having attended the seminars actively and passed the control test.
Examination has a practical and a theoretical part. In the practical part student has to illustrate the given tasks on particular examples.
Theoretical part includes questions related to the subject-matter presented at the lectures.
Controlled participation in lessons:
Absence has to be made up by self-study using recommended literature.
Type of course unit:
    Lecture  13 × 2 hrs. optionally                  
    Exercise  13 × 1 hrs. compulsory                  
Course curriculum:
    Lecture 1. Space of integrable functions - definition and basic properties, dense subsets,
convergence theorems.
2. Space of quadratically integrable functions - different kinds of convergence, Fourier series.
3. Singular integral - definition, representation, application to Fourier series.
4. Trigonometric series.
5. Fourier integral.
6. Fourier transformation - Fourier transformation (FT), inverse formula, basic properties of FT, Hermit and Laguer functions, FT and convolution, applications.
7. Plancherel theorem, Hermit functions.
8. Laplace transformation.
    Exercise 1. Space of integrable functions - definition and basic properties, dense subsets, convergence theorems.
2. Space of quadratically integrable functions - different kinds of convergence, Fourier series.
3. Singular integral - definition, representation, application to Fourier series.
4. Trigonometric series.
5. Fourier integral.
6. Fourier transformation - Fourier transformation (FT), inverse formula, basic properties of FT, Hermit and Laguer functions, FT and convolution, applications.
7. Plancherel theorem, Hermit functions.
8. Laplace transformation
Literature - fundamental:
1. I. P. Natanson: Teorija funkcij veščestvennoj peremennoj, [Theory of functions of a real variable] ,Third edition, "Nauka'', Moscow, 1974.
2. A. N. Kolmogorov, S. V. Fomin: Základy teorie funkcí a funkcionální analýzy, SNTL, Praha 1975.
3. E. W. Howel, B. Keneth: Principles of Fourier Analysis, CRC Press, 2001.
4. E. M. Stein´, G. Weiss: Introduction to Fourier Analysis on Eucledian spaces, Princeton University Press, 1971.
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
M2A-P full-time study M-MAI Mathematical Engineering -- GCr 4 Compulsory 2 1 S