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.
