Definitions

# Trigonometric series

In mathematics, a trigonometric series is any series of the form:

$frac\left\{1\right\}\left\{2\right\}A_\left\{o\right\}+displaystylesum_\left\{n=1\right\}^\left\{infty\right\}\left(A_\left\{n\right\} cos\left\{nx\right\} + B_\left\{n\right\} sin\left\{nx\right\}\right).$

It is called a Fourier series when the terms $A_\left\{n\right\}$ and $B_\left\{n\right\}$ have the form:

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

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

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

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