1. One-dimensional Fourier transform

At the very beginning of the nineteenth century, J. Fourier, far from Paris and the world of science, devoted some of his spare time as the (very active) Prefect of Isère to solving the equations of heat propagation in a solid. To facilitate study, he represented the function he was looking for as a sum of sinusoids, whose derivation gives a function of the same type. He also found the formula for calculating the amplitude of each sinusoid in this decomposition: projection onto a basis of orthogonal functions.

At the same time and independently (although his writings on this subject were not published until much later), C. F. Gauss was studying the trajectories of celestial bodies. To interpolate their functions between observation times, he used periodic functions and their decomposition into harmonics; to speed...