240034 Proseminar Fourier Series (PS) (SoSe 2020)

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Content of the course:

1] Basic properties and first applications of Fourier series:

Motivations, Fourier coefficients and Fourier series, convolution kernels, notions of convergence, applications to the isoperimetric inequality and the vibrating string.

2] The Fourier transform on the line and applications:

Definition for integrable functions, Schwartz space, Fourier inversion, Plancherel formula, Poisson summation formula, applications to the heat equation.

3] Finite Fourier analysis:

Modular arithmetic, properties of Fourier series on Z/nZ, Fast Fourier Transform, generalization to finite abelian groups, application to arithmetic progressions.

Requirements for participation, required level

This seminar will be taught in English. This is a good opportunity to start broadening the literature to English written textbooks. A very basic knowledge of English is enough, as the focus is foremost on the mathematical content.

Bibliography

E. M. Stein, R. Shakarchi, "Fourier analysis: an introduction", Princeton University Press (2011).
H. Dym, H. P. McKean, "Fourier series and integrals", Probability and Mathematical Statistics, Academic Press, New York-London (1972).