In mathematics, the Fejér kernel is a summability kernel used to express the effect of Cesàro summation on Fourier series. It is a non-negative kernel, giving rise to an approximate identity. It is named after the Hungarian mathematician Lipót Fejér (1880–1959).
The Fejér kernel is defined as
F
n
(
x
)
=
1
n
∑
k
=
0
n
−
1
D
k
(
x
)
,
where
D
k
(
x
)
=
∑
s
=
−
k
k
e
i
s
x
is the kth order Dirichlet kernel. It can also be written in a closed form as
F
n
(
x
)
=
1
n
(
sin
n
x
2
sin
x
2
)
2
=
1
n
(
1
−
cos
(
n
x
)
1
−
cos
x
)
,
where this expression is defined.
The Fejér kernel can also be expressed as
F
n
(
x
)
=
∑
|
j
|
≤
n
(
1
−
|
j
|
n
)
e
i
j
x
.
The Fejér kernel is a positive summability kernel. An important property of the Fejér kernel is
F
n
(
x
)
≥
0
with average value of
1
.
The convolution Fn is positive: for
f
≥
0
of period
2
π
it satisfies
0
≤
(
f
∗
F
n
)
(
x
)
=
1
2
π
∫
−
π
π
f
(
y
)
F
n
(
x
−
y
)
d
y
.
Since
f
∗
D
n
=
S
n
(
f
)
=
∑
|
j
|
≤
n
f
^
j
e
i
j
x
, we have
f
∗
F
n
=
1
n
∑
k
=
0
n
−
1
S
k
(
f
)
, which is Cesàro summation of Fourier series.
By Young's inequality,
∥
F
n
∗
f
∥
L
p
(
[
−
π
,
π
]
)
≤
∥
f
∥
L
p
(
[
−
π
,
π
]
)
for every
1
≤
p
≤
∞
for
f
∈
L
p
.
Additionally, if
f
∈
L
1
(
[
−
π
,
π
]
)
, then
f
∗
F
n
→
f
a.e.
Since
[
−
π
,
π
]
is finite,
L
1
(
[
−
π
,
π
]
)
⊃
L
2
(
[
−
π
,
π
]
)
⊃
⋯
⊃
L
∞
(
[
−
π
,
π
]
)
, so the result holds for other
L
p
spaces,
p
≥
1
as well.
If
f
is continuous, then the convergence is uniform, yielding a proof of the Weierstrass theorem.
One consequence of the pointwise a.e. convergence is the uniquess of Fourier coefficients: If
f
,
g
∈
L
1
with
f
^
=
g
^
, then
f
=
g
a.e. This follows from writing
f
∗
F
n
=
∑
|
j
|
≤
n
(
1
−
|
j
|
n
)
f
^
j
e
i
j
t
, which depends only on the Fourier coefficients.
A second consequence is that if
lim
n
→
∞
S
n
(
f
)
exists a.e., then
lim
n
→
∞
F
n
(
f
)
=
f
a.e., since Cesàro means
F
n
∗
f
converge to the original sequence limit if it exists.
