Fejér's theorem

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. ${\displaystyle 2\pi }$

It was proven by Fejér in 1900.

statement

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: ${\displaystyle {\mathcal {C}}_{2\pi }:=\left\{f\in C(\mathbb {R} );\,f(x)=f(x+2\pi )\;\forall \,x\in \mathbb {R} \right\}}$ ${\displaystyle 2\pi }$ ${\displaystyle n}$ ${\displaystyle s_{n}\;(n\in \mathbb {N} )}$ ${\displaystyle f\in {\mathcal {C}}_{2\pi }}$ ${\displaystyle \textstyle s_{n}(x):=\sum _{k=-n}^{n}c_{k}e^{ikx}}$ ${\displaystyle \textstyle c_{k}:={\frac {1}{2\pi }}\int _{0}^{2\pi }f(x)e^{-ikx}\mathrm {d} x}$

Be , then converge ${\displaystyle f\in {\mathcal {C}}_{2\pi }}$

${\displaystyle {\frac {1}{n+1}}\sum _{k=0}^{n}s_{k}(x)}$

for evenly in against . ${\displaystyle n\rightarrow \infty }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle f(x)}$

annotation

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. ${\displaystyle f\in {\mathcal {C}}_{2\pi }}$
• 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. ${\displaystyle {\mathcal {C}}_{2\pi }}$

• The Fourier series expansion is clear: two functions from have the same Fourier series if and only if they agree as functions. ${\displaystyle {\mathcal {C}}_{2\pi }}$

• The partial sums of a function converge in the -norm to the function, ie for , where ${\displaystyle f\in {\mathcal {C}}_{2\pi }}$ ${\displaystyle L_{2\pi }^{2}}$ ${\displaystyle \lim _{n\rightarrow \infty }\left\|f-s_{n}\right\|_{L_{2\pi }^{2}}=0}$ ${\displaystyle \left\|g\right\|_{L_{2\pi }^{2}}:=\left({\frac {1}{2\pi }}\int _{0}^{2\pi }\left|g(x)\right|^{2}\mathrm {d} x\right)^{1/2}}$

• For applies the so-called Bessel equation : wherein the Fourier coefficients of are. ${\displaystyle f\in {\mathcal {C}}_{2\pi }}$ ${\displaystyle \left\|f\right\|_{L_{2\pi }^{2}}^{2}=\sum _{k=-\infty }^{\infty }\left|c_{k}\right|^{2}}$ ${\displaystyle c_{k}}$ ${\displaystyle f}$

• By polarizing one obtains Parseval's theorem from the Bessel equation : Let with Fourier coefficients or . Then :, where is the L ² scalar product. ${\displaystyle f,g\in {\mathcal {C}}_{2\pi }}$ ${\displaystyle c_{k}}$ ${\displaystyle d_{k}}$ ${\displaystyle \langle f,g\rangle =\sum _{k=-\infty }^{\infty }c_{k}{\overline {d_{k}}}}$ ${\displaystyle \langle f,g\rangle ={\frac {1}{2\pi }}\int _{0}^{2\pi }f(x)\cdot {\overline {g(x)}}\,\mathrm {d} x}$

See also

• Fejér polynomials

literature

• Konrad Königsberger : Analysis 1 . Springer, Berlin 2004, ISBN 3-540-41282-4

Individual evidence

1. ^ Fejér, Sur les fonctions bornées et intégrables, Comptes Rendus Acad. Sci. Paris, Volume 131, 1900, pp. 984-987