# Sergei Yakovenko's blog: on Math and Teaching

## Tuesday, January 26, 2016

### Lecture 12, Jan 21, 2016

Filed under: lecture,Rothschild Course "Symmetry" — Sergei Yakovenko @ 4:51
Tags: ,

# Basics of the Fourier series

1. Low regularity functions: example, discussion. Stock market, home electricity meter…
2. Euclidean geometry in the $n$-space, real and complex. Normed bases and their advantage. Projections.
3. Scalar product on the space of functions as the limit of finite-dimensional scalar products.
4. Fourier coefficients in the real and complex form. Rate of their decay for functions of different regularity.
5. Fourier series and their main properties (without proofs). Completeness of the exponential/trigonometric system, Parseval equality.
6. Connection with holomorphic functions. Converging power series as a source of fast converging Fourier series.