In mathematics, Fejér's theorem,[1][2] named after Hungarian mathematician Lipót Fejér, states the following:[3] Fejér's Theorem — Let
be the nth partial sum of the Fourier series of
Explicitly, we can write the Fourier series of f as
where the nth partial sum of the Fourier series of f may be written as where the Fourier coefficients
are Then, we can define with Fn being the nth order Fejér kernel.
With the convergence written out explicitly, the above statement becomes
We first prove the following lemma: Lemma 1 — The nth partial sum of the Fourier series
may be written using the Dirichlet Kernel as:
We substitute the integral form of the Fourier coefficients into the formula for
{\displaystyle s_{n}(f,x)=\sum _{k=-n}^{n}c_{k}e^{ikx}=\sum _{k=-n}^{n}[{\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(t)e^{-ikt}dt]e^{ikx}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(t)\sum _{k=-n}^{n}e^{ik(x-t)}\,dt={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(t)\,D_{n}(x-t)\,dt.}
may be written using the Fejér Kernel as:
Proof: Recall the definition of the Fejér Kernel
As in the case of Lemma 1, we substitute the integral form of the Fourier coefficients into the formula for
is a geometric sum, we get an simple formula for
We are now ready to prove Fejér's Theorem.
First, let us recall the statement we are trying to prove
We begin by invoking Lemma 2:
Applying the triangle inequality yields
We first note that the function f is continuous on [-π,π].
We invoke the theorem that every periodic function on [-π,π] that is continuous is also bounded and uniformily continuous.
This gives the desired bound for integral 1 which we can exploit in final step.
By Lemma 3c we know that the integral goes to 0 as n goes to infinity, and because epsilon is arbitrary, we can set it equal to 0.
In fact, Fejér's theorem can be modified to hold for pointwise convergence.
converges pointwise as n goes to infinity.
Sadly however, the theorem does not work in a general sense when we replace the sequence
This is because there exist functions whose Fourier series fails to converge at some point.
This fact, called Lusins conjecture or Carleson's theorem, was proven in 1966 by L.
A more general form of the theorem applies to functions which are not necessarily continuous (Zygmund 1968, Theorem III.3.4).
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 (Zygmund 1968, Theorem III.5.1).