Plan of lectures

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

Part 1. Fourier series 

 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.

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

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

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

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

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

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

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