Difference between revisions of "Fejér's theorem"

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In mathematics, '''Fejér's theorem''', named for [[Hungary|Hungarian]] [[mathematician]] [[Lipót Fejér]], states that if ''f'':'''R'''&nbsp;→&nbsp;'''C''' is a [[continuous function]] with [[periodic function|period]] 2π, then the [[sequence]] (σ<sub>''n''</sub>) of [[Cesàro mean]]s of the sequence (''s''<sub>''n''</sub>) of [[partial sum]]s of the [[Fourier series]] of ''f'' [[Uniform convergence|converges uniformly]] to ''f'' on [-π,π].  
In mathematics, '''Fejér's theorem''', named for [[Hungary|Hungarian]] [[mathematician]] [[Lipót Fejér]], states that if ''f'':'''R'''&nbsp;→&nbsp;'''C''' is a [[continuous function]] with [[periodic function|period]] 2π, then the [[sequence]] (σ<sub>''n''</sub>) of [[Cesàro mean]]s of the sequence (''s''<sub>''n''</sub>) of [[partial sum]]s of the [[Fourier series]] of ''f'' [[Uniform convergence|converges uniformly]] to ''f'' on [-π,π].


Explicitly,
Explicitly,
:<math>s_n(x)=\sum_{k=-n}^nc_ke^{ikx},</math>
:<math>s_n(x)=\sum_{k=-n}^nc_ke^{ikx},</math>
where
where
:<math>c_n=\frac{1}{2\pi}\int_{-\pi}^\pi f(t)e^{-int}dt,</math>
:<math>c_k=\frac{1}{2\pi}\int_{-\pi}^\pi f(t)e^{-ikt}dt,</math>
and
and
:<math>\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,</math>
:<math>\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,</math>
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* {{citation|title=Trigonometric series|first=Antoni|last=Zygmund|authorlink=Antoni Zygmund|publisher=Cambridge University Press|year=1968|publication-date=1988|isbn=978-0-521-35885-9|edition=2nd}}.
* {{citation|title=Trigonometric series|first=Antoni|last=Zygmund|authorlink=Antoni Zygmund|publisher=Cambridge University Press|year=1968|publication-date=1988|isbn=978-0-521-35885-9|edition=2nd}}.


{{DEFAULTSORT:Fejer's theorem}}
[[Category:Fourier series]]
[[Category:Fourier series]]
[[Category:Theorems in approximation theory]]
[[Category:Theorems in approximation theory]]
[[de:Satz von Fejér]]
[[fr:Théorème de Fejér]]
[[pl:Twierdzenie Fejéra]]

Latest revision as of 09:59, 20 August 2013

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,

where

and

with Fn being the nth order Fejér kernel.

A more general form of the theorem applies to functions which are not necessarily continuous Template:Harv. 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

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 Template:Harv.

References

  • {{#invoke:citation/CS1|citation

|CitationClass=citation }}.