Course detail

# Fourier Analysis

FSI-SFACompulsoryMaster's (2nd cycle)Acad. year: 2016/2017Winter semester1. year of study4  credits

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.

Learning outcomes of the course unit

Knowledge of basic topics of Fourier Analysis, manely, Fourier series, Fourier and Laplace transformations, and ability to apply this knowledge in practice.

Mode of delivery

90 % face-to-face, 10 % distance learning

Prerequisites

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

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Recommended or required reading

I. P. Natanson: Teorija funkcij veščestvennoj peremennoj, Nauka, Moskva, 1954.
A. N. Kolmogorov, S. V. Fomin: Základy teorie funkcí a funkcionální analýzy, SNTL, Praha 1975.
E. W. Howel, B. Keneth: Principles of Fourier Analysis, CRC Press, 2001.
E. M. Stein´, G. Weiss: Introduction to Fourier Analysis on Eucledian spaces, Princeton University Press, 1971.

Planned learning activities and teaching methods

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

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.

Language of instruction

Czech

Work placements

Not applicable.

Course curriculum

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.

Aims

The aim of the course is to familiarise students with basic topics and techniques of the Fourier analysis used in other mathematical subjects

Specification of controlled education, way of implementation and compensation for absences

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

#### Type of course unit

Lecture

26 hours, optionally

Teacher / Lecturer

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.

seminars

13 hours, compulsory

Teacher / Lecturer