Besov space

In mathematics, the Besov space (named after Oleg Vladimirovich Besov) $B_{p,q}^{s}(\mathbf {R} )$ is a complete quasinormed space which is a Banach space when 1 ≤ p, q ≤ ∞. It, as well as the similarly defined Triebel–Lizorkin space, serve to generalize more elementary function spaces and are effective at measuring regularity properties of functions.

Definition

Several equivalent definitions exist. One of them is described below.

Let

$\Delta _{h}f(x)=f(x-h)-f(x)$

and define the modulus of continuity by

$\omega _{p}^{2}(f,t)=\sup _{|h|\leq t}\left\|\Delta _{h}^{2}f\right\|_{p}$

Let n be a non-negative integer and define: s = n + α with 0 < α ≤ 1. The Besov space $B_{p,q}^{s}(\mathbf {R} )$ contains all functions f such that (see Sobolev space)

$f\in W^{n,p}(\mathbf {R} ),\qquad \int _{0}^{\infty }\left|{\frac {\omega _{p}^{2}\left(f^{(n)},t\right)}{t^{\alpha }}}\right|^{q}{\frac {dt}{t}}<\infty .$

Norm

The Besov space $B_{p,q}^{s}(\mathbf {R} )$ is equipped with the norm

$\left\|f\right\|_{B_{p,q}^{s}(\mathbf {R} )}=\left(\|f\|_{W^{n,p}(\mathbf {R} )}^{q}+\int _{0}^{\infty }\left|{\frac {\omega _{p}^{2}\left(f^{(n)},t\right)}{t^{\alpha }}}\right|^{q}{\frac {dt}{t}}\right)^{\frac {1}{q}}$

The Besov spaces $B_{2,2}^{s}(\mathbf {R} )$ coincide with the more classical Sobolev spaces $H^{s}(\mathbf {R} )$.

If $p=q$ and $s$ is not an integer, then $B_{p,p}^{s}(\mathbf {R} )={\bar {W}}^{s,p}(\mathbf {R} )$, where ${\bar {W}}^{s,p}(\mathbf {R} )$ denotes the Sobolev–Slobodeckij space.

References

• Triebel, H. "Theory of Function Spaces II".
• Besov, O. V. "On a certain family of functional spaces. Embedding and extension theorems", Dokl. Akad. Nauk SSSR 126 (1959), 1163–1165.
• 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).