Distributions and Fourier Transform

Introduction. The theory of distributions, or generalized functions, provides a unified framework for performing standard calculus operations on nonsmooth functions, measures (such as the Dirac delta function), and even more general measure-like objects in the same way as they are done for smooth functions. In this theory, any distribution can be differentiated arbitrarily many times, a large class of distributions have well-defined Fourier transforms, and general linear operators can be expressed as integral operators with distributional kernel. The distributional point of view is very useful since it easily allows to perform such operations in a certain weak sense. However, often additional work is required if stronger statements are needed. The theory in its modern form arose from the work of Laurent Schwartz in the late 1940s, although it certainly had important precursors such as Heaviside’s operational calculus in the 1890s and Sobolev’s generalized functions in the 1930s. The approach of Schwartz had the important feature of being completely mathematically rigorous while retaining the ease of calculation of the operational methods. Distributions have played a prominent role in the modern theory of partial differential equations, and they will be used heavily in the chapter on Microlocal methods in this encyclopedia. The idea behing distribution theory is easily illustrated by the standard example, the Dirac delta function. On the real line, the Dirac delta is a ”function δ(x) which is zero for x 6= 0 with an infinitely high peak at x = 0, with area equal to one”. Thus, if f(x) is a smooth function then integrating δ(x)f(x) is supposed to give ∫ ∞