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 TheoremMay 10. Lecture 7. Plancherel theorem for finite commutative groups¶

May 17. Lecture 8. reserve. TBD¶

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