Trigonometric series.html

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

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

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

$A_{n}=\frac{1}{\pi}\displaystyle\int^{2 \pi}_0\! f(x) \cos{nx} \,dx\qquad (n=0,1,2, \dots)$

$B_{n}=\frac{1}{\pi}\displaystyle\int^{2 \pi}_0\! f(x) \sin{nx}\, dx\qquad (n=1,2,3, \dots)$

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

^ ^a ^b Fourier Series and Orthogonal Functions By Harry F. Davis. Page 89

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