In mathematics, a summability kernel is a family or sequence of periodic integrable functions satisfying a certain set of properties, listed below. Certain kernels, such as the Fejér kernel, are particularly useful in Fourier analysis. Summability kernels are related to approximation of the identity; definitions of an approximation of identity vary, but sometimes the definition of an approximation of the identity is taken to be the same as for a summability kernel.
Let
T
:=
R
/
Z
. A summability kernel is a sequence
(
k
n
)
in
L
1
(
T
)
that satisfies
-
∫
T
k
n
(
t
)
d
t
=
1
-
∫
T
|
k
n
(
t
)
|
d
t
≤
M
(uniformly bounded)
-
∫
δ
≤
|
t
|
≤
1
2
|
k
n
(
t
)
|
d
t
→
0
as
n
→
∞
, for every
δ
>
0
.
Note that if
k
n
≥
0
for all
n
, i.e.
(
k
n
)
is a positive summability kernel, then the second requirement follows automatically from the first.
If instead we take the convention
T
=
R
/
2
π
Z
, the first equation becomes
1
2
π
∫
T
k
n
(
t
)
d
t
=
1
, and the upper limit of integration on the third equation should be extended to
π
.
We can also consider
R
rather than
T
; then we integrate (1) and (2) over
R
, and (3) over
|
t
|
>
δ
.
The Fejér kernel
The Poisson kernel (continuous index)
The Dirichlet kernel is not a summability kernel, since it fails the second requirement.
Let
(
k
n
)
be a summability kernel, and
∗
denote the convolution operation.
If
(
k
n
)
,
f
∈
C
(
T
)
(continuous functions on
T
), then
k
n
∗
f
→
f
in
C
(
T
)
, i.e. uniformly, as
n
→
∞
.
If
(
k
n
)
,
f
∈
L
1
(
T
)
, then
k
n
∗
f
→
f
in
L
1
(
T
)
, as
n
→
∞
.
If
(
k
n
)
is radially decreasing symmetric and
f
∈
L
1
(
T
)
, then
k
n
∗
f
→
f
pointwise a.e., as
n
→
∞
. This uses the Hardy–Littlewood maximal function. If
(
k
n
)
is not radially decreasing symmetric, but the decreasing symmetrization
k
~
n
(
x
)
:=
sup
|
y
|
≥
|
x
|
k
n
(
y
)
satisfies
sup
n
∈
N
∥
k
n
~
∥
1
<
∞
, then a.e. convergence still holds, using a similar argument.
