Plan of lectures

The plan of lectures given below is of preliminary nature. It may be changed during the course.

Part 1. Fourier series (lecturer: Serguei Shimorin)

19 January. Lecture 1. Unit circle S^1 as a group. Characters. Orthogonal series of exponents in L^2(S^1). Sections 4.1, 4.2, 1.3.

26 January. Lecture 2. L^2(0,1) as a Hilbert space. Properties of Fourier series in L^2(S^1). Sections 1.2, 1.3.

2 February. Lecture 3. Fourier series of smooth and continuous functions. Dirichlet and Fejer kernels. Section 1.4.

9 February. Lecture 4. Fourier series of L^1 functions. Convolutions and Fourier series. Section 1.5.

16 February.Lecture 5. Applications of Fourier series: isoperimetric inequality; polynomial approximation. Section 1.7.

23 February. Lecture 6. One-dimensional heat equation and string equation. Section 1.8.

1 March. Lecture 7. Hardy functions in the unit disk. Section 3.8.

8 March. Lecture 8. Dirichlet problem for Laplacian in the disk (lecture notes). Several-dimensional Fourier series. Section 1.10.  Fourier series and harmonic function in the disk

Part 2. Fourier integrals (lecturer: Maurice Duits)

March 29. Lecture 1. Fourierintegrals for rapidly decreasing functions and on  L1 

April 5. Lecture 2.  Fourierintegrals  on L2. Plancherel Theorem. Hermite Functions. Several Dimensions.

April 12. Lecture 3. Applications: Heat equation, Heisenberg’s inequality, Poisson Summation.

April 19. Lecture 4. Function Theory, Phragmén-Lindelöf, Hardy Theorem, Paley Wiener Theorem.

April 26. Lecture 5. Function Theory continued.

May 3. Lecture 6. Applications: Prime number theorem, Szasz-Müntz Theorem

May 10. Lecture 7. Plancherel theorem for finite commutative groups

May 17. Lecture 8. reserve. TBD