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.