$frac\{1\}\{2\}A\_\{o\}+displaystylesum\_\{n=1\}^\{infty\}(A\_\{n\}\; cos\{nx\}\; +\; B\_\{n\}\; sin\{nx\}).$

It is called a Fourier series when the terms $A\_\{n\}$ and $B\_\{n\}$ have the form:

$A\_\{n\}=frac\{1\}\{pi\}displaystyleint^\{2\; pi\}\_0!\; f(x)\; cos\{nx\}\; ,dxqquad\; (n=0,1,2,\; dots)$

$B\_\{n\}=frac\{1\}\{pi\}displaystyleint^\{2\; pi\}\_0!\; f(x)\; sin\{nx\},\; dxqquad\; (n=1,2,3,\; dots)$

where $f$ is an integrable function.

It is not that case that every trigonometric series is a Fourier Series. A particular question of interest is given a trigonometric series, for which values of x does the series converge.

References

• "Trigonmetric Series" by A. Zygmund