Harmonic analysis, distributions, convolution

Harmonic analysis is originally the branch of mathematics that deals with periodic or quasi-periodic signals (with a definition that we'll define in the course of this article). Introduced by Fourier for the study of the heat equation, where he achieved great success, it very quickly became an essential tool not only for mathematicians (for the resolution of certain equations, such as wave equations or convolution equations), but also for physicists (for wave phenomena or propagation, optics, etc.), astronomers (for the study of the effects of light on the Earth's surface) and astronomers.), astronomers (celestial mechanics, spectroscopy), electricians (electrical circuit equations) and even music (since sounds are precisely periodic sound signals), from which it derives its harmonic attribute. These applications have lost none of their importance, but they have been augmented by many others since we generalized the concept of Fourier series decomposition, applicable only to periodic functions, into a Fourier transformation, usable on a far greater number of functions.

The basic ideas of harmonic analysis are very simple, and can essentially be summed up in this profession of faith: reduce everything to basic functions whose properties are well known (sine and cosine functions, or exponentials), by expressing "general functions" in the form of sums, or more generally integrals, of such "elementary functions". But their practical application poses a number of difficulties, both theoretical (what exactly is a "general function"?) and practical (how can such a decomposition be achieved, or on the contrary how can the function be recomposed from its expression in these elementary functions; what are the properties of the decomposed image of a function, etc.). These problems have been much debated by mathematicians since the last century, but it's only fairly recently that a fully satisfactory solution has been found, providing an elementary and general framework for the Fourier transformation (and many other mathematical questions besides): the theory of distributions, conceived by L. Schwartz in the 1950s. We'll begin by outlining its main elements, rather like setting the scene for a play. We'll also mention the important concept of convolution of two functions or two distributions, which plays an essential role in electronics and optics, for example.

In the third section, we explain the concept of the Fourier transform and its usual properties. Fourier series, although historically older, will only be explained in the fourth section, as their properties follow very simply from those of the Fourier transform.

The practical calculation of Fourier transforms, or Fourier series, is covered in the fifth section. There are a number of algebraic...