Fejér's theorem
In mathematics, the set of Fejér (after Leopold Fejér ) is one of the most important statements on the convergence of Fourier series. The theorem states that the arithmetic mean of the partial sums of the Fourier series of a continuous, periodic function converge uniformly to the function.
Statement
Be the space of continuous periodic functions. The - th partial sum of the Fourier series of a feature is given by the Fourier coefficients. The set of Fejér is now:
Be, then converges
For evenly against.
Note
The set of Fejér can not be further tightened in this form:
- Leopold Fejér constructed in 1911 an example of a function whose Fourier series does not converge in at least one point.
- If the condition of continuity weakened to piecewise continuity, converge, the arithmetical mean of the partial sums in the discontinuity no longer against the function value.
Consequences
- If a Fourier series of a function to converge at a point, then they will converge to the function value.
- The Fourier series expansion is clear: Two functions have exactly the same then the Fourier series if they coincide as functions.
- The partial sums of a function converge in the norm against the function, ie, with
- For the so-called Bessel equation :, where the Fourier coefficients of are.
- By polarization is obtained from the Bessel equation the set of Parseval: Be with Fourier coefficients or. Then: where the L2 - scalar product is.