Historical Background

Unarguably, modern analysis was formed during the resolution of an important controversy (or, rather, controversies) concerning the representation of “arbitrary” functions. This controversy has unfolded slowly over the last two centuries and was put to its final rest only in our own time.

The story begins in 1746 with the famous vibrating string problem. Briefly, an elastic string of length L has each end fastened to one of the endpoints of the interval [0, L] on the x-axis and is set into motion (as you might pluck a guitar string, for example). The problem is to determine the position y = F(x, t) of the string at time t, given only its initial position y = f(x) = F(x, 0) at time t = 0 where, for simplicity, we assume that the initial velocity Ft(x, 0) = 0. The function F(x, t) is the solution to d'Alembert's wave equation: Ftt = a2Fxx, where a is a positive constant determined by certain physical properties of the string. The initial data for the problem is F(x, 0) = f(x), Ft(x, 0) = 0, and f(0) = 0 = f(L).

The controversy, initially between d'Alembert and Euler, centers around the nature of the functions f that may be permitted as initial positions.