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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].

Explicitly,
:<math>s_n(x)=\sum_{k=-n}^nc_ke^{ikx},</math>
where
:<math>c_k=\frac{1}{2\pi}\int_{-\pi}^\pi f(t)e^{-ikt}dt,</math>
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>
with ''F''<sub>''n''</sub> being the ''n''th order [[Fejér kernel]].

A more general form of the theorem applies to functions which are not necessarily continuous {{harv|Zygmund|1968|loc=Theorem III.3.4}}.  Suppose that ''f'' is in ''L''<sup>1</sup>(-π,π).  If the left and right limits ''f''(''x''<sub>0</sub>±0) of ''f''(''x'') exist at ''x''<sub>0</sub>, or if both limits are infinite of the same sign, then

:<math>\sigma_n(x_0) \to \frac{1}{2}\left(f(x_0+0)+f(x_0-0)\right).</math>

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 σ<sub>''n''</sub> is replaced with [[Cesàro summation|(C, &alpha;) mean]] of the Fourier series {{harv|Zygmund|1968|loc=Theorem III.5.1}}.

==References==
* {{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:Theorems in approximation theory]]