trigonometric series

A trigonometric series is any series of the form a_0/2 + sum of (a_n cos(n x) + b_n sin(n x)) with arbitrary real (or complex) coefficients — no function f is presumed to exist behind it. Think of it as a blueprint for a wave built from harmonics, written down before we ask whether the wave it describes is a sensible function at all.

Every Fourier series is a trigonometric series, but not conversely. The Fourier series of an integrable f is the special trigonometric series whose coefficients arise as integrals of f. There exist trigonometric series that converge everywhere yet are the Fourier series of no integrable function — for instance sum of (sin(n x))/log(n) for n >= 2 converges for every x but its ‘coefficients’ b_n = 1/log(n) are not square-summable, so it cannot be the Fourier series of any L2 function.

The distinction launched the subject. Cantor's study of when a trigonometric series can equal zero everywhere (the question of uniqueness) led him to construct the real numbers and to invent set theory. So this seemingly narrow notion is, historically, the seed from which much of modern analysis and set theory grew.