Trigonometric series

In mathematics, a trigonometric series is an infinite series of the form


 * $$ A_0 + \sum_{n=1}^\infty A_n \cos{nx} + B_n \sin{nx},$$

where $$x$$ is the variable and $$\{A_n\}$$ and $$\{B_n\}$$ are coefficients. It is an infinite version of a trigonometric polynomial.

A trigonometric series is called the Fourier series of the integrable function $f$ if the coefficients have the form:


 * $$A_n=\frac1\pi \int^{2 \pi}_0\! f(x) \cos{nx} \,dx$$


 * $$B_n=\frac{1}{\pi}\displaystyle\int^{2 \pi}_0\! f(x) \sin{nx}\, dx$$

Examples
Every Fourier series gives an example of a trigonometric series. Let the function $$f(x) = x$$ on $$[-\pi,\pi]$$ be extended periodically (see sawtooth wave). Then its Fourier coefficients are:
 * $$\begin{align}

A_n &= \frac1\pi\int_{-\pi}^{\pi} x \cos{nx}\,dx = 0, \quad n \ge 0. \\[4pt] B_n &= \frac1\pi\int_{-\pi}^{\pi} x \sin{nx}\, dx \\[4pt] &= -\frac{x}{n\pi} \cos{nx} + \frac1{n^2\pi}\sin{nx} \Bigg\vert_{x=-\pi}^\pi \\[5mu] &= \frac{2\,(-1)^{n+1}}{n}, \quad n \ge 1.\end{align}$$ Which gives an example of a trigonometric series:
 * $$2\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} \sin{nx} = 2\sin{x} - \frac22\sin{2x} + \frac23\sin{3x} - \frac24\sin{4x} + \cdots$$

The converse is false however, not every trigonometric series is a Fourier series. The series
 * $$\sum_{n=2}^\infty \frac{\sin{nx}}{\log{n}} = \frac{\sin{2x}}{\log{2}} + \frac{\sin{3x}}{\log{3}} + \frac{\sin{4x}}{\log{4}}+\cdots$$

is a trigonometric series which converges for all $$x$$ but is not a Fourier series. Here $$B_n=\frac{1}{\log(n)}$$ for $$n\geq 2$$ and all other coefficients are zero.

Uniqueness of Trigonometric series
The uniqueness and the zeros of trigonometric series was an active area of research in 19th century Europe. First, Georg Cantor proved that if a trigonometric series is convergent to a function $$f(x)$$ on the interval $$[0, 2\pi]$$, which is identically zero, or more generally, is nonzero on at most finitely many points, then the coefficients of the series are all zero.

Later Cantor proved that even if the set S on which $$f$$ is nonzero is infinite, but the derived set S' of S is finite, then the coefficients are all zero. In fact, he proved a more general result. Let S0 = S and let Sk+1 be the derived set of Sk. If there is a finite number n for which Sn is finite, then all the coefficients are zero. Later, Lebesgue proved that if there is a countably infinite ordinal α such that Sα is finite, then the coefficients of the series are all zero. Cantor's work on the uniqueness problem famously led him to invent transfinite ordinal numbers, which appeared as the subscripts α in Sα.