# Besov space

In mathematics, the **Besov space** (named after Oleg Vladimirovich Besov) is a complete quasinormed space which is a Banach space when 1 ≤ *p*, *q* ≤ ∞. These spaces, as well as the similarly defined Triebel–Lizorkin spaces, serve to generalize more elementary function spaces such as Sobolev spaces and are effective at measuring regularity properties of functions.

## Definition

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

Let

and define the modulus of continuity by

Let n be a non-negative integer and define: *s* = *n* + *α* with 0 < *α* ≤ 1. The Besov space contains all functions f such that

## Norm

The Besov space is equipped with the norm

The Besov spaces coincide with the more classical Sobolev spaces .

If and is not an integer, then , where 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).