Course detail
Fourier Analysis
FSI-SFA-AAcad. year: 2020/2021
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.
Supervisor
Department
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.
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, [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. M. Stein´, G. Weiss: Introduction to Fourier Analysis on Eucledian spaces, Princeton University Press, 1971 (EN)
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)
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
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.
Language of instruction
English
Work placements
Not applicable.
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.
Exercise
13 hours, compulsory
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