In mathematics, the Besov space (named after Oleg Vladimirovich Besov)
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.
Several equivalent definitions exist. One of them is given below.
Let
![{\displaystyle \Delta _{h}f(x)=f(x-h)-f(x)}](//wikimedia.org/api/rest_v1/media/math/render/svg/9327f37c47f7fec0f03ec864cbb65b1036d8ebb7)
and define the modulus of continuity by
![{\displaystyle \omega _{p}^{2}(f,t)=\sup _{|h|\leq t}\left\|\Delta _{h}^{2}f\right\|_{p}}](//wikimedia.org/api/rest_v1/media/math/render/svg/a24c2e1befe5ed9139338ed9c36608457ddc625f)
Let n be a non-negative integer and define: s = n + α with 0 < α ≤ 1. The Besov space
contains all functions f such that
![{\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 .}](//wikimedia.org/api/rest_v1/media/math/render/svg/9fed86c740608fb6e7f691bd863de3f56062f2d7)
The Besov space
is equipped with the norm
![{\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}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/d222faf9223fbe761541ffcb48c9095ad5ca0b8c)
The Besov spaces
coincide with the more classical Sobolev spaces
.
If
and
is not an integer, then
, where
denotes the Sobolev–Slobodeckij space.
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- 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