# Basics of the Fourier series

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

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