# Besov space

In mathematics, the Besov space (named after Oleg Vladimirovich Besov) $B^s_{p,q}(\mathbf{R})$ 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 (in a sense) smoothness properties of functions.

## Definition

Let

$\Delta_h f(x) = f(x-h) - f(x)$

and the modulus of continuity is defined by

$\omega^2_p(f,t) = \sup_{|h| \le t} \left \| \Delta^2_h f \right \|_p$

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

$f \in W^{n, p}(\mathbf{R}), \qquad \int_0^\infty \left|\frac{ \omega^2_p \left ( f^{(n)},t \right ) } {t^{\alpha} }\right|^q \frac{dt}{t} < \infty.$

## Norm

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

$\left \|f \right \|_{B^s_{p,q}(\mathbf{R})} = \left( \|f\|_{W^{n, p} (\mathbf{R})}^q + \int_0^\infty \left|\frac{ \omega^2_p \left ( f^{(n)}, t \right ) } {t^{\alpha} }\right|^q \frac{dt}{t} \right)^{\frac{1}{q}}$

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

## 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.
• Weisstein, Eric W. "Besov Space." From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/BesovSpace.html
• DeVore, R., Kyriazis, G. and Wang, P. "Multiscale characterizations of Besov spaces on bounded domains", Journal of Approximation Theory 93, 273-292 (1998).