Definition
edit
The Fejér kernel has many equivalent definitions. We outline three such definitions below:
1) The traditional definition expresses the Fejér kernel
F
n
(
x
)
{\displaystyle F_{n}(x)}
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 .
2) The Fejér kernel
F
n
(
x
)
{\displaystyle F_{n}(x)}
[1]
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}}\left({\frac {1-\cos(nx)}{1-\cos(x)}}\right)}
This closed form expression may be derived from the definitions used above. The proof of this result goes as follows.
First, we use the fact that the Dirichlet kernel may be written as:[2]
D
k
(
x
)
=
sin
(
k
+
1
2
)
x
sin
x
2
{\displaystyle D_{k}(x)={\frac {\sin(k+{\frac {1}{2}})x}{\sin {\frac {x}{2}}}}}
Hence, using the definition of the Fejér kernel above we get:
F
n
(
x
)
=
1
n
∑
k
=
0
n
−
1
D
k
(
x
)
=
1
n
∑
k
=
0
n
−
1
sin
(
(
k
+
1
2
)
x
)
sin
(
x
2
)
=
1
n
1
sin
(
x
2
)
∑
k
=
0
n
−
1
sin
(
(
k
+
1
2
)
x
)
=
1
n
1
sin
2
(
x
2
)
∑
k
=
0
n
−
1
[
sin
(
(
k
+
1
2
)
x
)
⋅
sin
(
x
2
)
]
{\displaystyle F_{n}(x)={\frac {1}{n}}\sum _{k=0}^{n-1}D_{k}(x)={\frac {1}{n}}\sum _{k=0}^{n-1}{\frac {\sin((k+{\frac {1}{2}})x)}{\sin({\frac {x}{2}})}}={\frac {1}{n}}{\frac {1}{\sin({\frac {x}{2}})}}\sum _{k=0}^{n-1}\sin((k+{\frac {1}{2}})x)={\frac {1}{n}}{\frac {1}{\sin ^{2}({\frac {x}{2}})}}\sum _{k=0}^{n-1}[\sin((k+{\frac {1}{2}})x)\cdot \sin({\frac {x}{2}})]}
Using the trigonometric identity:
sin
(
α
)
⋅
sin
(
β
)
=
1
2
(
cos
(
α
−
β
)
−
cos
(
α
+
β
)
)
{\displaystyle \sin(\alpha )\cdot \sin(\beta )={\frac {1}{2}}(\cos(\alpha -\beta )-\cos(\alpha +\beta ))}
F
n
(
x
)
=
1
n
1
sin
2
(
x
2
)
∑
k
=
0
n
−
1
[
sin
(
(
k
+
1
2
)
x
)
⋅
sin
(
x
2
)
]
=
1
n
1
2
sin
2
(
x
2
)
∑
k
=
0
n
−
1
[
cos
(
k
x
)
−
cos
(
(
k
+
1
)
x
)
]
{\displaystyle F_{n}(x)={\frac {1}{n}}{\frac {1}{\sin ^{2}({\frac {x}{2}})}}\sum _{k=0}^{n-1}[\sin((k+{\frac {1}{2}})x)\cdot \sin({\frac {x}{2}})]={\frac {1}{n}}{\frac {1}{2\sin ^{2}({\frac {x}{2}})}}\sum _{k=0}^{n-1}[\cos(kx)-\cos((k+1)x)]}
Hence it follows that:
F
n
(
x
)
=
1
n
1
sin
2
(
x
2
)
1
−
cos
(
n
x
)
2
=
1
n
1
sin
2
(
x
2
)
sin
2
(
n
x
2
)
=
1
n
(
sin
(
n
x
2
)
sin
(
x
2
)
)
2
{\displaystyle F_{n}(x)={\frac {1}{n}}{\frac {1}{\sin ^{2}({\frac {x}{2}})}}{\frac {1-\cos(nx)}{2}}={\frac {1}{n}}{\frac {1}{\sin ^{2}({\frac {x}{2}})}}\sin ^{2}({\frac {nx}{2}})={\frac {1}{n}}({\frac {\sin({\frac {nx}{2}})}{\sin({\frac {x}{2}})}})^{2}}
3) The Fejér kernel can also be expressed as:
F
n
(
x
)
=
∑
|
k
|
≤
n
−
1
(
1
−
|
k
|
n
)
e
i
k
x
{\displaystyle F_{n}(x)=\sum _{|k|\leq n-1}\left(1-{\frac {|k|}{n}}\right)e^{ikx}}
Properties
edit
The Fejér kernel is a positive summability kernel. An important property of the Fejér kernel is
F
n
(
x
)
≥
0
{\displaystyle F_{n}(x)\geq 0}
1
{\displaystyle 1}
Convolution
edit
The convolution Fn is positive: for
f
≥
0
{\displaystyle f\geq 0}
2
π
{\displaystyle 2\pi }
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.}
Since
f
∗
D
n
=
S
n
(
f
)
=
∑
|
j
|
≤
n
f
^
j
e
i
j
x
{\displaystyle f*D_{n}=S_{n}(f)=\sum _{|j|\leq n}{\widehat {f}}_{j}e^{ijx}}
f
∗
F
n
=
1
n
∑
k
=
0
n
−
1
S
k
(
f
)
{\displaystyle f*F_{n}={\frac {1}{n}}\sum _{k=0}^{n-1}S_{k}(f)}
Cesàro summation of Fourier series.
By Young's convolution inequality ,
‖
F
n
∗
f
‖
L
p
(
[
−
π
,
π
]
)
≤
‖
f
‖
L
p
(
[
−
π
,
π
]
)
for every
1
≤
p
≤
∞
for
f
∈
L
p
.
{\displaystyle \|F_{n}*f\|_{L^{p}([-\pi ,\pi ])}\leq \|f\|_{L^{p}([-\pi ,\pi ])}{\text{ for every }}1\leq p\leq \infty {\text{ for }}f\in L^{p}.}
Additionally, if
f
∈
L
1
(
[
−
π
,
π
]
)
{\displaystyle f\in L^{1}([-\pi ,\pi ])}
f
∗
F
n
→
f
{\displaystyle f*F_{n}\rightarrow f}
a.e. Since
[
−
π
,
π
]
{\displaystyle [-\pi ,\pi ]}
L
1
(
[
−
π
,
π
]
)
⊃
L
2
(
[
−
π
,
π
]
)
⊃
⋯
⊃
L
∞
(
[
−
π
,
π
]
)
{\displaystyle L^{1}([-\pi ,\pi ])\supset L^{2}([-\pi ,\pi ])\supset \cdots \supset L^{\infty }([-\pi ,\pi ])}
L
p
{\displaystyle L^{p}}
p
≥
1
{\displaystyle p\geq 1}
If
f
{\displaystyle f}
Weierstrass theorem .
One consequence of the pointwise a.e. convergence is the uniqueness of Fourier coefficients: If
f
,
g
∈
L
1
{\displaystyle f,g\in L^{1}}
f
^
=
g
^
{\displaystyle {\hat {f}}={\hat {g}}}
f
=
g
{\displaystyle f=g}
f
∗
F
n
=
∑
|
j
|
≤
n
(
1
−
|
j
|
n
)
f
^
j
e
i
j
t
{\displaystyle f*F_{n}=\sum _{|j|\leq n}\left(1-{\frac {|j|}{n}}\right){\hat {f}}_{j}e^{ijt}}
A second consequence is that if
lim
n
→
∞
S
n
(
f
)
{\displaystyle \lim _{n\to \infty }S_{n}(f)}
lim
n
→
∞
F
n
(
f
)
=
f
{\displaystyle \lim _{n\to \infty }F_{n}(f)=f}
F
n
∗
f
{\displaystyle F_{n}*f}
See also
edit
References
edit
^ Hoffman, Kenneth (1988). Banach Spaces of Analytic Functions . Dover. p. 17. ISBN 0-486-45874-1 ^ Konigsberger, Konrad. Analysis 1 (in German) (6th ed.). Springer. p. 322. 
![{\displaystyle F_{n}(x)={\frac {1}{n}}\sum _{k=0}^{n-1}D_{k}(x)={\frac {1}{n}}\sum _{k=0}^{n-1}{\frac {\sin((k+{\frac {1}{2}})x)}{\sin({\frac {x}{2}})}}={\frac {1}{n}}{\frac {1}{\sin({\frac {x}{2}})}}\sum _{k=0}^{n-1}\sin((k+{\frac {1}{2}})x)={\frac {1}{n}}{\frac {1}{\sin ^{2}({\frac {x}{2}})}}\sum _{k=0}^{n-1}[\sin((k+{\frac {1}{2}})x)\cdot \sin({\frac {x}{2}})]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98fb8fd21819dce9f17372c5dfb57842f15a0c70)
![{\displaystyle F_{n}(x)={\frac {1}{n}}{\frac {1}{\sin ^{2}({\frac {x}{2}})}}\sum _{k=0}^{n-1}[\sin((k+{\frac {1}{2}})x)\cdot \sin({\frac {x}{2}})]={\frac {1}{n}}{\frac {1}{2\sin ^{2}({\frac {x}{2}})}}\sum _{k=0}^{n-1}[\cos(kx)-\cos((k+1)x)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f50481bb0f23b4e469b56513fb9bd47373f5cd1)
![{\displaystyle \|F_{n}*f\|_{L^{p}([-\pi ,\pi ])}\leq \|f\|_{L^{p}([-\pi ,\pi ])}{\text{ for every }}1\leq p\leq \infty {\text{ for }}f\in L^{p}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/292178857a312a42f0d6a0042c83b5e64fb17b4e)
![{\displaystyle f\in L^{1}([-\pi ,\pi ])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e3515a00783d75e503eb19294f26d84c49402fe)
![{\displaystyle [-\pi ,\pi ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb064fd6c55820cfa660eabeeda0f6e3c4935ae6)
![{\displaystyle L^{1}([-\pi ,\pi ])\supset L^{2}([-\pi ,\pi ])\supset \cdots \supset L^{\infty }([-\pi ,\pi ])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46f6aab539f73c629502ceebd84a0f671e9c0178)
