Course detail
Fourier Analysis
FSI-SFA-A Acad. year: 2023/2024 Summer semester
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
Supervisor
Department
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.
The study programmes with the given course
Programme N-MAI-A: Mathematical Engineering, Master's, compulsory
Programme N-AIM-A: Applied and Interdisciplinary Mathematics, Master's, compulsory
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