Fejér's theorem

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In mathematics, Fejér's theorem, named for Hungarian mathematician Lipót Fejér, states that if f:R  C is a continuous function with period 2π, then the sequencen) of Cesàro means of the sequence (sn) of partial sums of the Fourier series of f converges uniformly to f on [-π,π].

Explicitly,

s_{n}(x)=\sum _{{k=-n}}^{n}c_{k}e^{{ikx}},

where

c_{k}={\frac  {1}{2\pi }}\int _{{-\pi }}^{\pi }f(t)e^{{-ikt}}dt,

and

\sigma _{n}(x)={\frac  {1}{n}}\sum _{{k=0}}^{{n-1}}s_{k}(x)={\frac  {1}{2\pi }}\int _{{-\pi }}^{\pi }f(x-t)F_{n}(t)dt,

with Fn being the nth order Fejér kernel.

A more general form of the theorem applies to functions which are not necessarily continuous (Zygmund 1968, Theorem III.3.4). Suppose that f is in L1(-π,π). If the left and right limits f(x0±0) of f(x) exist at x0, or if both limits are infinite of the same sign, then

\sigma _{n}(x_{0})\to {\frac  {1}{2}}\left(f(x_{0}+0)+f(x_{0}-0)\right).

Existence or divergence to infinity of the Cesàro mean is also implied. By a theorem of Marcel Riesz, Fejér's theorem holds precisely as stated if the (C, 1) mean σn is replaced with (C, α) mean of the Fourier series (Zygmund 1968, Theorem III.5.1).

References

  • Zygmund, Antoni (1968), Trigonometric series (2nd ed.), Cambridge University Press (published 1988), ISBN 978-0-521-35885-9 .
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