In mathematics , the Fejér kernel is used to express the effect of Cesàro summation on Fourier series . It is a non-negative kernel, giving rise to an approximate identity .
Plot of several Fejér kernels
The Fejér kernel is defined as
F
n
(
x
)
=
1
n
∑
k
=
0
n
−
1
D
k
(
x
)
,
{\displaystyle F_{n}(x)={\frac {1}{n}}\sum _{k=0}^{n-1}D_{k}(x),}
where
D
k
(
x
)
=
∑
s
=
−
k
k
e
i
s
x
{\displaystyle D_{k}(x)=\sum _{s=-k}^{k}{\rm {e}}^{isx}}
is the k th 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
{\displaystyle F_{n}(x)={\frac {1}{n}}\left({\frac {\sin {\frac {nx}{2}}}{\sin {\frac {x}{2}}}}\right)^{2}={\frac {1}{n}}{\frac {1-\cos(nx)}{1-\cos x}}}
,
where this expression is defined.[1] It is named after the Hungarian mathematician Lipót Fejér (1880–1959).
The important property of the Fejér kernel is
F
n
(
x
)
≥
0
{\displaystyle F_{n}(x)\geq 0}
with average value of
1
{\displaystyle 1}
. The convolution Fn is positive: for
f
≥
0
{\displaystyle f\geq 0}
of period
2
π
{\displaystyle 2\pi }
it satisfies
0
≤
(
f
∗
F
n
)
(
x
)
=
1
2
π
∫
−
π
π
f
(
y
)
F
n
(
x
−
y
)
d
y
,
{\displaystyle 0\leq (f*F_{n})(x)={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(y)F_{n}(x-y)\,dy,}
and, by Young's inequality ,
‖
F
n
∗
f
‖
L
p
(
[
−
π
,
π
]
)
≤
‖
f
‖
L
p
(
[
−
π
,
π
]
)
{\displaystyle \|F_{n}*f\|_{L^{p}([-\pi ,\pi ])}\leq \|f\|_{L^{p}([-\pi ,\pi ])}}
for every
0
≤
p
≤
∞
{\displaystyle 0\leq p\leq \infty }
for continuous function
f
{\displaystyle f}
; moreover,
f
∗
F
n
→
f
{\displaystyle f*F_{n}\rightarrow f}
for every
f
∈
L
p
(
[
−
π
,
π
]
)
{\displaystyle f\in L^{p}([-\pi ,\pi ])}
(
1
≤
p
<
∞
{\displaystyle 1\leq p<\infty }
)
for continuous function
f
{\displaystyle f}
. Indeed, if
f
{\displaystyle f}
is continuous, then the convergence is uniform.
See also
References
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