In mathematics , Fejér's theorem (after Leopold Fejér ) is one of the most important statements about the convergence of Fourier series . The theorem says that the arithmetic means of the partial sums of the Fourier series of a continuous , periodic function converge uniformly to the function.
Let be the space of continuous -periodic functions. The -th partial sum of the Fourier series of a function is given by with the Fourier coefficients . Fejér's theorem now reads:
Be , then converge
for evenly in against .
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Fejér's theorem cannot be further tightened in this form:
In 1911 Leopold Fejér constructed an example of a function whose Fourier series does not converge in at least one point.
If the condition of continuity is weakened to piecewise continuity, the arithmetic means of the partial sums in the discontinuities no longer converge to the function value.
Consequences
If a Fourier series of a function from converges at a point, then it converges to the function value.
The Fourier series expansion is clear: two functions from have the same Fourier series if and only if they agree as functions.
The partial sums of a function converge in the -norm to the function, ie for , where
For applies the so-called Bessel equation : wherein the Fourier coefficients of are.
By polarizing one obtains Parseval's theorem from the Bessel equation : Let with Fourier coefficients or . Then :, where is the L 2 scalar product.