As was the case with the Dirichlet kernel, the Fejer kernel has some properties.
Lemma 11.6.3The Fejer kernel has the following properties.
F_{n+1}
(t)
= F_{n+1}
(t+ 2π)
∫_{−π}^{π}F_{n+1}
(t)
dt = 1
∫_{−π}^{π}F_{n+1}
(t)
f
(x − t)
dt = ∑_{k=−n}^{n}b_{k}e^{ikθ}for a suitable choice of b_{k}.
F_{n+1}
(t)
=
1−cos((n+1)tt)-
4π(n+1)sin2(2)
, F_{n+1}
(t)
≥ 0,F_{n}
(t)
= F_{n}
(− t)
.
For every δ > 0,
nl→im∞ sup {Fn+1(t) : π ≥ |t| ≥ δ} = 0.
In fact, for
|t|
≥ δ,
2
Fn+1 (t) ≤ --------2(δ)---.
(n + 1)sin 2 4π
Proof: Part 1.) is obvious because F_{n+1} is the average of functions for which this is
true.
Part 2.) is also obvious for the same reason as Part 1.). Part 3.) is obvious because it is
true for D_{n} in place of F_{n+1} and then taking the average yields the same sort of
sum.
The last statements in 4.) are obvious from the formula which is the only hard part of
4.).
(( ) )
-------1-------∑n 1
Fn+1 (t) = (n + 1)sin(t) 2π sin k+ 2 t
2 k=0
n∑ ( ( ) ) ( )
= -------12-(t)--- sin k + 1 t sin t
(n+ 1)sin 2 2π k=0 2 2
Here is a picture of the Fejer kernels for n=2,4,6.
PICT
Note how these kernels are nonnegative, unlike the Dirichlet kernels. Also there is a large
bump in the center which gets increasingly large as n gets larger. The fact these kernels are
nonnegative is what is responsible for the superior ability of the Fejer means to approximate a
continuous function.
Theorem 11.6.4Let f be a continuous and 2π periodic function. Then
lim ||f − σ f||= 0.
n→∞ n+1 0
Proof: Let ε > 0 be given. Then by part 2. of Lemma 11.6.3,
Now choose δ such that for all x, it follows that if
|y|
< δ then
|f (x) − f (x− y)| < ε∕2.
This can be done because f is uniformly continuous on
[− π,π]
by Theorem 6.7.2 on Page
290. Since it is periodic, it must also be uniformly continuous on ℝ. (why?) Therefore, for this
δ, this has shown that for all x