Course detail

Fourier Analysis

FSI-SFA-AAcad. year: 2023/2024

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.

Language of instruction

English

Number of ECTS credits

4

Mode of study

Not applicable.

Guarantor

prof. Aleksandre Lomtatidze, DrSc.

Department

Institute of Mathematics (ÚM)

Entry knowledge

Calculus, basic konwledge of linear functional analysis, measure theory.

Rules for evaluation and completion of the course

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.
Absence has to be made up by self-study using recommended literature.

Aims

The aim of the course is to familiarise students with basic topics and techniques of the Fourier analysis used in other mathematical subjects
Knowledge of basic topics of Fourier Analysis, manely, Fourier series, Fourier and Laplace transformations, and ability to apply this knowledge in practice.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

I. P. Natanson: Teorija funkcij veščestvennoj peremennoj, [Theory of functions of a real variable] ,Third edition, "Nauka'', Moscow, 1974. (RU)
A. N. Kolmogorov, S. V. Fomin: Základy teorie funkcí a funkcionální analýzy, SNTL, Praha 1975. (CS)
E. W. Howel, B. Keneth: Principles of Fourier Analysis, CRC Press, 2001. (EN)
L. Grafakos, Classical Fourier Analysis: Third edition, Graduate Texts in Mathematics, 249. Springer, New York, 2014. (EN)

Recommended reading

E. M. Stein´, G. Weiss: Introduction to Fourier Analysis on Eucledian spaces, Princeton University Press, 1971 (EN)

Classification of course in study plans

  • Programme N-MAI-A Master's, 1. year of study, summer semester, compulsory
  • Programme N-AIM-A Master's, 2. year of study, summer semester, compulsory

Type of course unit

 

Lecture

26 hours, optionally

Teacher / Lecturer

prof. Aleksandre Lomtatidze, DrSc.

Syllabus

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

13 hours, compulsory

Teacher / Lecturer

prof. Aleksandre Lomtatidze, DrSc.

Syllabus

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