Moduli of smoothness
edit
The modulus of smoothness of order
n
{\displaystyle n}
[ 1]
f
∈
C
[
a
,
b
]
{\displaystyle f\in C[a,b]}
ω
n
:
[
0
,
∞
)
→
R
{\displaystyle \omega _{n}:[0,\infty )\to \mathbb {R} }
ω
n
(
t
,
f
,
[
a
,
b
]
)
=
sup
h
∈
[
0
,
t
]
sup
x
∈
[
a
,
b
−
n
h
]
|
Δ
h
n
(
f
,
x
)
|
for
0
≤
t
≤
b
−
a
n
,
{\displaystyle \omega _{n}(t,f,[a,b])=\sup _{h\in [0,t]}\sup _{x\in [a,b-nh]}\left|\Delta _{h}^{n}(f,x)\right|\qquad {\text{for}}\quad 0\leq t\leq {\frac {b-a}{n}},}
and
ω
n
(
t
,
f
,
[
a
,
b
]
)
=
ω
n
(
b
−
a
n
,
f
,
[
a
,
b
]
)
for
t
>
b
−
a
n
,
{\displaystyle \omega _{n}(t,f,[a,b])=\omega _{n}\left({\frac {b-a}{n}},f,[a,b]\right)\qquad {\text{for}}\quad t>{\frac {b-a}{n}},}
where the finite difference (n -th order forward difference) is defined as
Δ
h
n
(
f
,
x
0
)
=
∑
i
=
0
n
(
−
1
)
n
−
i
(
n
i
)
f
(
x
0
+
i
h
)
.
{\displaystyle \Delta _{h}^{n}(f,x_{0})=\sum _{i=0}^{n}(-1)^{n-i}{\binom {n}{i}}f(x_{0}+ih).}
1.
ω
n
(
0
)
=
0
,
ω
n
(
0
+
)
=
0.
{\displaystyle \omega _{n}(0)=0,\omega _{n}(0+)=0.}
2.
ω
n
{\displaystyle \omega _{n}}
[
0
,
∞
)
.
{\displaystyle [0,\infty ).}
3.
ω
n
{\displaystyle \omega _{n}}
[
0
,
∞
)
.
{\displaystyle [0,\infty ).}
4. For
m
∈
N
,
t
≥
0
{\displaystyle m\in \mathbb {N} ,t\geq 0}
ω
n
(
m
t
)
≤
m
n
ω
n
(
t
)
.
{\displaystyle \omega _{n}(mt)\leq m^{n}\omega _{n}(t).}
5.
ω
n
(
f
,
λ
t
)
≤
(
λ
+
1
)
n
ω
n
(
f
,
t
)
,
{\displaystyle \omega _{n}(f,\lambda t)\leq (\lambda +1)^{n}\omega _{n}(f,t),}
λ
>
0.
{\displaystyle \lambda >0.}
6. For
r
∈
N
{\displaystyle r\in \mathbb {N} }
W
r
{\displaystyle W^{r}}
[
−
1
,
1
]
{\displaystyle [-1,1]}
(
r
−
1
)
{\displaystyle (r-1)}
[
−
1
,
1
]
{\displaystyle [-1,1]}
‖
f
(
r
)
‖
L
∞
[
−
1
,
1
]
<
+
∞
.
{\displaystyle \left\|f^{(r)}\right\|_{L_{\infty }[-1,1]}<+\infty .}
If
f
∈
W
r
,
{\displaystyle f\in W^{r},}
ω
r
(
t
,
f
,
[
−
1
,
1
]
)
≤
t
r
‖
f
(
r
)
‖
L
∞
[
−
1
,
1
]
,
t
≥
0
,
{\displaystyle \omega _{r}(t,f,[-1,1])\leq t^{r}\left\|f^{(r)}\right\|_{L_{\infty }[-1,1]},t\geq 0,}
where
‖
g
(
x
)
‖
L
∞
[
−
1
,
1
]
=
e
s
s
sup
x
∈
[
−
1
,
1
]
|
g
(
x
)
|
.
{\displaystyle \|g(x)\|_{L_{\infty }[-1,1]}={\mathrm {ess} \sup }_{x\in [-1,1]}|g(x)|.}
Moduli of smoothness can be used to prove estimates on the error of approximation. Due to property (6), moduli of smoothness provide more general estimates than the estimates in terms of derivatives.
For example, moduli of smoothness are used in Whitney inequality to estimate the error of local polynomial approximation. Another application is given by the following more general version of Jackson inequality :
For every natural number
n
{\displaystyle n}
f
{\displaystyle f}
2
π
{\displaystyle 2\pi }
trigonometric polynomial
T
n
{\displaystyle T_{n}}
≤
n
{\displaystyle \leq n}
|
f
(
x
)
−
T
n
(
x
)
|
≤
c
(
k
)
ω
k
(
1
n
,
f
)
,
x
∈
[
0
,
2
π
]
,
{\displaystyle \left|f(x)-T_{n}(x\right)|\leq c(k)\omega _{k}\left({\frac {1}{n}},f\right),\quad x\in [0,2\pi ],}
where the constant
c
(
k
)
{\displaystyle c(k)}
k
∈
N
.
{\displaystyle k\in \mathbb {N} .}
^ DeVore, Ronald A., Lorentz, George G., Constructive approximation, Springer-Verlag, 1993.
![{\displaystyle f\in C[a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18a75183a73efed5373e1d10284bceb0c4e48741)
![{\displaystyle \omega _{n}(t,f,[a,b])=\sup _{h\in [0,t]}\sup _{x\in [a,b-nh]}\left|\Delta _{h}^{n}(f,x)\right|\qquad {\text{for}}\quad 0\leq t\leq {\frac {b-a}{n}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50826e822a583c0b8a606be96d49cbe08af2f8fa)
![{\displaystyle \omega _{n}(t,f,[a,b])=\omega _{n}\left({\frac {b-a}{n}},f,[a,b]\right)\qquad {\text{for}}\quad t>{\frac {b-a}{n}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d3f65f4d187d6fcb7d1aa64bf431a5f50a57afd)
![{\displaystyle [-1,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51e3b7f14a6f70e614728c583409a0b9a8b9de01)
![{\displaystyle \left\|f^{(r)}\right\|_{L_{\infty }[-1,1]}<+\infty .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1da9c131d4bfdf03135bed2e89bc62fe30b2e77a)
![{\displaystyle \omega _{r}(t,f,[-1,1])\leq t^{r}\left\|f^{(r)}\right\|_{L_{\infty }[-1,1]},t\geq 0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c82b2530ae6caa586c542f801cb9c297d2dd2351)
![{\displaystyle \|g(x)\|_{L_{\infty }[-1,1]}={\mathrm {ess} \sup }_{x\in [-1,1]}|g(x)|.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1aa5745ad3bb73c7ec5b7c53cbe4541a9c6c6de2)
![{\displaystyle \left|f(x)-T_{n}(x\right)|\leq c(k)\omega _{k}\left({\frac {1}{n}},f\right),\quad x\in [0,2\pi ],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba122e08f91037ab2904a580b1d290473fd0a951)
