Trigonometric Series

Abstract

The functions represented by power series, or as Lagrange called them, the “analytic functions,” play indeed a central role in analysis. But the class of analytic functions is too restricted in many instances. It was therefore an event of major importance for all of mathematics and for a great variety of applications when Fourier in his “Théorie analytique de la chaleur”¹ observed and illustrated by many examples the fact that convergent trigonometric series of the form

$$f(x)=\frac{{{a_0}}}{2}+\sum\limits_{v = 1}^\infty {({a_v}\cos{\text{}}vx+{b_v}\sin{\text{}}vx)}$$

(1)

with constant coefficients a _v , b _v are capable of representing a wide class of “arbitrary” functions f(x), a class which includes essentially every function of specific interest, whether defined geometrically by mechanical means, or in any other way: even functions possessing jump discontinuities, or obeying different laws of formation in different intervals, can thus be expressed.