Optical heterodyne detection

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

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

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

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

The Besov space $B_{p,q}^{s}(\mathbb {R} )$ is equipped with the norm $\|f\|_{B_{p,q}^{s}(\mathbb {R} )}=\left(\|f\|_{W_{p}^{n}(\mathbb {R} )}^{q}+\int _{0}^{\infty }\left|{\frac {\omega _{p}^{2}(f^{(n)},t)}{t^{\alpha }}}\right|^{q}{\frac {dt}{t}}\right)^{1/q}$

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

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).