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Real Fourier-Series

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The real Fourier series of a real $ 2\pi$-periodic function $ f$ is an orthogonal expansion in terms of sines and cosines:

$\displaystyle f(x) \sim \frac{a_0}{2} + \sum_{k=1}^\infty \left( a_k\cos(kx)+b_k\sin(kx)\right) $

with

$\displaystyle a_k$ $\displaystyle = \frac{1}{\pi} \int\limits_{-\pi}^\pi f(t)\cos(kt)\,dt,\quad k\ge0,$    
$\displaystyle b_k$ $\displaystyle = \frac{1}{\pi} \int\limits_{-\pi}^\pi f(t)\sin(kt)\,dt,\quad k\ge1\,.$    

The type and rate of convergence of the series depends on the smoothness of $ f$. For example, sufficient for absolute convergence is that the series $ \sum_{k=0}^\infty \vert a_k\vert$ and $ \sum_{k=1}^\infty \vert b_k\vert$ converge.

A convergent Fourier series needs not to represent the true function value at every point. Usually, at a point of discontinuity, the series converges to the arithmetic mean of the left and right limits of $ f$. This explains the notation $ f(x) \sim \sum \cdots$ instead of $ f(x)= \sum \cdots$, which emphasizes these convergence issues.

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  automatically generated 9/22/2016