Distributions - Operations and derivatives

Many physical, chemical, biological and even economic phenomena can be modeled by differential equations or partial differential equations. The solution of a differential equation is an n-fold continuously differentiable function. However, at the beginning of the 20th century, it became apparent that these differentiability constraints were too restrictive and that – for certain phenomena – it could be interesting to introduce discontinuous functions as solutions. In the 1930s, Jean Leray introduced the notion of weak solutions for hydrodynamic equations (turbulent solutions of the Navier-Stokes equations). Shortly afterwards, Leonid Sobolev applied this concept to potential theory. Building on this work and seeking to give it a coherent framework, Laurent Schwartz (1945-1950) developed a general and rigorous theory known as "distribution theory".

At the same time, since the end of the 19th century, Heaviside's symbolic calculus had been a real success with engineers, because, although it often defied the mathematical rules in use, it had the merit of leading to exact results.

Then, in 1926, Dirac introduced his famous function, zero outside the origin, worth + ∞ at the origin and with an integral equal to 1, to model a unit impulse at time t = 0 and with zero effect outside t = 0. Even more difficult for the mathematician to admit, this δ function was also introduced as the derivative of the Heaviside H function, i.e. the function that is 1 for x > 0 and 0 for x < 0, a function that is precisely not derivable in 0! Dirac's function was used in integration-by-parts calculations, it was derived (referred to as δ ′), it was assigned a Fourier transform worth 1, convolution products were manipulated.

Once again, these various operations make perfect sense in the context of distribution theory.

In this study, we present the entire "toolbox" of this theory. But before reaching the purely operative aspect, the introduction of the main objects requires working in functional spaces, sometimes unfamiliar to the non-mathematician. After all, working in functional spaces is unavoidable when trying to solve partial differential equations!

This first article introduces the main operations on distributions and discusses the fundamental notion of the derivative of a distribution.

A second article will focus on the convolution product of distributions and their Fourier transform.

We've kept the use of fine topology properties to a...