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Fourier series

In mathematics, a Fourier series, named in honor of Joseph Fourier, is a representation of a periodic function as a sum of periodic functions of the form

<math>x\mapsto e^{inx},</math>
which are harmonics of a fundamental. Suppose f(x) is a complex-valued function of a real number, is periodic with period 2π, and is square integrable over the interval from 0 to 2π. Let

<math>\hat{f}(n)=\frac{1}{2\pi}\int_{-\pi}^\pi f(x)\,e^{-inx}\,dx.</math>

Then the Fourier series representation of f(x) is given by

<math> f(x)=\sum_{n=-\infty}^{\infty} \hat{f}(n)\,e^{inx} .</math>

Since

<math>e^{inx}=\cos(nx)+i\sin(nx)</math>
this is equivalent to representing f(x) as a infinite linear combination of functions of the form <math>\cos(nx)\quad{\rm and }\sin(nx)</math>, i.e.
<math>f(x) = \frac{1}{2}a_0 + \sum_{n=1}^\infty\left[a_n\cos(nx)+b_n\sin(n)\right], \quad{\rm where}</math>
<math>a_n = \frac{1}{\pi}\int_{-\pi}^\pi f(x)\cos(nx)dx\quad{\rm and}
\quad{\rm }b_n = \frac{1}{\pi}\int_{-\pi}^\pi f(x)\sin(nx)dx</math>

Does this series actually converge to f(x)?

A partial answer is that if f is square-integrable then

<math>\lim_{N\rightarrow\infty}\int_{-\pi}^\pi\left|f(x)-\sum_{n=-N}^{N} \hat{f}(n)\,e^{inx}\right|^2\,dx=0.</math>

That much was proved in the 19th century, as was the fact that if f is piecewise continuous[?] then the series converges at each point of continuity. Perhaps surprisingly, it was not shown until the 1960s that if f is quadratically integrable then the series converges for every value of x except those in some set of measure zero.

See also: Fourier transform, harmonic analysis.



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