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 $1\le p,q\le \mathrm{\infty }.$ 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

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

and the modulus of continuity is defined by

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

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

$f\in W_p^n(\mathbb{ℝ})$ and $\underset{0}{\overset{\mathrm{\infty }}{\int }}{\left|\frac{{\omega }_p^2(f^{(n)},t)}{t^{\alpha }}\right|}^q\frac{dt}{t}<\mathrm{\infty }$

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

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).