Definitions

trigonometric-series

Trigonometric series

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

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

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