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Fourier series and applications

Code: 36924
ECTS: 5.0
Lecturers in charge: izv. prof. dr. sc. Vjekoslav Kovač - Lectures
Lecturers: Mario Stipčić - Exercises
English level:

1,0,0

All teaching activities will be held in Croatian. However, foreign students in mixed groups will have the opportunity to attend additional office hours with the lecturer and teaching assistants in English to help master the course materials. Additionally, the lecturer will refer foreign students to the corresponding literature in English, as well as give them the possibility of taking the associated exams in English.
Load:

1. komponenta

Lecture typeTotal
Lectures 30
Exercises 30
* Load is given in academic hour (1 academic hour = 45 minutes)
Description:
COURSE AIMS AND OBJECTIVES: To introduce students to the basis of Fourier analysis and to provide a strong motivation for continuation of their study in this direction.

COURSE DESCRIPTION AND SYLLABUS:
1. Introduction. Normed spaces L1 and L2. Bases in L2.
2. Fourier coefficients. Fourier series. Complex form. Examples. L2 - convergence. Plancherel identity.
3. The question of pointwise convergence. Riemann - Lebesgue lemma.
4. Dirichlet kernel. Dirichlet integral. Riemann localization principle. Dini test. Lipschitz test.
5. Dirichlet theorem. Uniform convergence.
6. Gibbs phenomenon. Convergence discussion: Kolmogorov example, Carleson theorem.
7. Cesaro summability. Fejer theorem.
8. Various applications: Weierstrass aproximation, isoperimetric problem, heat equation.
9. Motivational lecture 1: Group structure and generalizations.
10. Motivational lecture 2: Fourier integral, applications to differential equations, applications to central limit theorem.
11. Motivational lecture 3: Heisenberg inequality, applications to information theory, Gabor systems.
12. Motivational lecture 4: Signal analysis and synthesis, wavelets.
13. Motivational lecture 5: Prime number theorem.
Literature:
5. semester Not active
Izborni predmet 1, 2 - Mandatory studij - Mathematics

6. semester
Izborni predmet 1, 2 - Mandatory studij - Mathematics
Consultations schedule: