Many problems associated with signal propagation or wave propagation can be essentially simplified by applying the Fourier transform to the transmitted function (by working in the frequency domain instead of the time domain).

I would like to provide few examples of Fourier transform applications to practical problems, including:
a) How the music for Tarkovsky's Solaris movie was recorded?
b) Why the first transatlantic telegraph cable was laid in 1858, and the first transatlantic telegraph telephone cable was laid in 1956 (almost 100 years later)?
c) Why inductive coils were inserted into telegraph cables (pupinization)?

1.  The Fourier transform. Fourier series and Fourier integral. Dini's criterion. Convergence of the Fourier series for Lipschitz functions. The Laplace transform.
2. The Fourier series, Fej\'er series and Vallée-Poussin series. The Gibbs phenomenon.
3. The Fourier transform as a unitary map in $L^2$. The Fourier transform in the spaces $L^p$.
4. The human ear as a Fourier analyzer. Which distortions in the audio equipment are treated as ``natural'' and which are irritating? What does it mean ``tone control''?
5. Fourier analysis and Fourier synthesis. The rainbow as an analog Fourier transform. Mursin's ANS synthesizer user for recording music for Tarkovsky's Solaris. Analog Fourier analyzer for Project Cyclops.
6. Signal transition through a linear ``black box''. Description in terms of the convolution operator. Action of the convolution operator in the Fourier representation.
7. Fourier transform representation in the radio engineering. Modulation and the spectral shift.
8. Why the first transatlantic telegraph cable was laid in 1858, and the first transatlantic telegraph telephone cable was laid in 1956 (almost 100 years later)? Telegraph equations. Birth of operational calculus. Dispersion control. Pupinization.
9. Discrete Fourier transform. Fast Fourier Transform. Applications in the digital signal processing. Digital compensation of the transmission lines imperfection.
10. The black hole in the center of our Galaxy. How such a sharp image was obtained? Synthetic-aperture radio telescopes.
11. Digital sound. Kotelnikov theorem. Multibit and 1-bit digital-analog converters.
12. Application of Fast Fourier Transform to numerical integration of non-linear partial differential equations. The split-step method. Connections with the Campbell–Hausdorff theorem.

Lecturer
Grinevich Petr Georgievich

Financial support
The course is supported by the Ministry of Science and Higher Education of the Russian Federation (the grant to the Steklov International Mathematical Center, agreement no. 075-15-2022-265).

Institutions
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center