##
Examples

The Fourier series for the identity function suffers from the

Gibbs phenomenon near the ends of the periodic interval.

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:

- ${\begin{aligned}A_{n}&={\frac {1}{\pi ))\int _{-\pi }^{\pi }x\cos {nx}\,dx=0,\quad n\geq 0.\\[4pt]B_{n}&={\frac {1}{\pi ))\int _{-\pi }^{\pi }x\sin {nx}\,dx\\[4pt]&=-{\frac {x}{n\pi ))\cos {nx}+{\frac {1}{n^{2}\pi ))\sin {nx}{\Bigg \vert }_{x=-\pi }^{\pi }\\[5mu]&={\frac {2\,(-1)^{n+1)){n)),\quad n\geq 1.\end{aligned))$

Which gives an example of a trigonometric series:

- $2\sum _{n=1}^{\infty }{\frac {(-1)^{n+1)){n))\sin {nx}=2\sin {x}-{\frac {2}{2))\sin {2x}+{\frac {2}{3))\sin {3x}-{\frac {2}{4))\sin {4x}+\cdots$

The trigonometric series sin 2*x* / log 2 + sin 3*x* / log 3 + sin 4*x* / log 4 + ... is not a Fourier series.

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.^{[1]}
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.^{[2]}

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 *S*_{0} = *S* and let *S*_{k+1} be the derived set of *S*_{k}. If there is a finite number *n* for which *S*_{n} is finite, then all the coefficients are zero. Later, Lebesgue proved that if there is a countably infinite ordinal *α* such that *S*_{α} 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 *S*_{α} .^{[3]}