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Generalization of Sobolev spaces
In
mathematics , the Besov space (named after
Oleg Vladimirovich Besov )
B
p
,
q
s
(
R
)
{\displaystyle B_{p,q}^{s}(\mathbf {R} )}
is a
complete
quasinormed space which is a
Banach space when 1 ≤ p , q ≤ ∞ . These spaces, as well as the similarly defined
Triebel–Lizorkin spaces , serve to generalize more elementary
function spaces such as
Sobolev spaces and are effective at measuring regularity properties of functions.
Definition
Several equivalent definitions exist. One of them is given below.
Let
Δ
h
f
(
x
)
=
f
(
x
−
h
)
−
f
(
x
)
{\displaystyle \Delta _{h}f(x)=f(x-h)-f(x)}
and define the
modulus of continuity by
ω
p
2
(
f
,
t
)
=
sup
|
h
|
≤
t
‖
Δ
h
2
f
‖
p
{\displaystyle \omega _{p}^{2}(f,t)=\sup _{|h|\leq t}\left\|\Delta _{h}^{2}f\right\|_{p}}
Let n be a non-negative integer and define: s = n + α with 0 < α ≤ 1 . The Besov space
B
p
,
q
s
(
R
)
{\displaystyle B_{p,q}^{s}(\mathbf {R} )}
contains all functions f such that
f
∈
W
n
,
p
(
R
)
,
∫
0
∞
|
ω
p
2
(
f
(
n
)
,
t
)
t
α
|
q
d
t
t
<
∞
.
{\displaystyle f\in W^{n,p}(\mathbf {R} ),\qquad \int _{0}^{\infty }\left|{\frac {\omega _{p}^{2}\left(f^{(n)},t\right)}{t^{\alpha }}}\right|^{q}{\frac {dt}{t}}<\infty .}
Norm
The Besov space
B
p
,
q
s
(
R
)
{\displaystyle B_{p,q}^{s}(\mathbf {R} )}
is equipped with the norm
‖
f
‖
B
p
,
q
s
(
R
)
=
(
‖
f
‖
W
n
,
p
(
R
)
q
+
∫
0
∞
|
ω
p
2
(
f
(
n
)
,
t
)
t
α
|
q
d
t
t
)
1
q
{\displaystyle \left\|f\right\|_{B_{p,q}^{s}(\mathbf {R} )}=\left(\|f\|_{W^{n,p}(\mathbf {R} )}^{q}+\int _{0}^{\infty }\left|{\frac {\omega _{p}^{2}\left(f^{(n)},t\right)}{t^{\alpha }}}\right|^{q}{\frac {dt}{t}}\right)^{\frac {1}{q}}}
The Besov spaces
B
2
,
2
s
(
R
)
{\displaystyle B_{2,2}^{s}(\mathbf {R} )}
coincide with the more classical
Sobolev spaces
H
s
(
R
)
{\displaystyle H^{s}(\mathbf {R} )}
.
If
p
=
q
{\displaystyle p=q}
and
s
{\displaystyle s}
is not an integer, then
B
p
,
p
s
(
R
)
=
W
¯
s
,
p
(
R
)
{\displaystyle B_{p,p}^{s}(\mathbf {R} )={\bar {W}}^{s,p}(\mathbf {R} )}
, where
W
¯
s
,
p
(
R
)
{\displaystyle {\bar {W}}^{s,p}(\mathbf {R} )}
denotes the
Sobolev–Slobodeckij space .
References
Triebel, Hans (1992). Theory of Function Spaces II .
doi :
10.1007/978-3-0346-0419-2 .
ISBN
978-3-0346-0418-5 .
Besov, O. V. (1959). "On some families of functional spaces. Imbedding and extension theorems". Dokl. Akad. Nauk SSSR (in Russian). 126 : 1163–1165.
MR
0107165 .
DeVore, R. and Lorentz, G. "Constructive Approximation", 1993.
DeVore, R., Kyriazis, G. and Wang, P. "Multiscale characterizations of Besov spaces on bounded domains", Journal of Approximation Theory 93, 273-292 (1998).
Leoni, Giovanni (2017).
A First Course in Sobolev Spaces: Second Edition .
Graduate Studies in Mathematics . 181 . American Mathematical Society. pp. 734.
ISBN
978-1-4704-2921-8
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