Besov space

In mathematics, the Besov space (named after Oleg Vladimirovich Besov) $B^s_{p,q}(\R)$ is a complete quasinormed space which is a Banach space when $1 \le p, q \le \infty.$ 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.

Let

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

and the modulus of continuity is defined by

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

Let $n=0,1,2,\ldots,\quad s=n+\alpha$ with $0<\alpha\le 1$, the Besov space $B^s_{p,q}(\R)$ contains all functions $f$ such that

$f \in W^n_p(\R)$ and $\int_{0}^\infty \left|\frac{ \omega^2_p ( f^{(n)},t) } {t^{\alpha} }\right|^q \frac{dt}{t} < \infty$

The Besov space $B^s_{p,q}(\R)$ is equipped with the norm $\|f\|_{B^s_{p,q}(\R)} = \left(\|f\|_{W^n_p(\R)}^q + \int_0^\infty \left|\frac{ \omega^2_p ( f^{(n)},t) } {t^{\alpha} }\right|^q \frac{dt}{t} \right)^{1/q}$

If $p=q=2$, the Besov spaces $B^s_{2,2}(\R)$ coincide with the more classical Sobolev spaces $H^s(\R)$.

References

• 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