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

Let

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

and define the modulus of continuity by

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

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

${\displaystyle f\in W^{n,p}(\mathbf {R} ),\qquad \int _{0}^{\infty }\left|{\frac {\omega _{p}^{2}\left(f^{(n)},t\right)}{t^{\alpha }}}\right|^{q}{\frac {dt}{t}}<\infty .}$ 

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

${\displaystyle \left\|f\right\|_{B_{p,q}^{s}(\mathbf {R} )}=\left(\|f\|_{W^{n,p}(\mathbf {R} )}^{q}+\int _{0}^{\infty }\left|{\frac {\omega _{p}^{2}\left(f^{(n)},t\right)}{t^{\alpha }}}\right|^{q}{\frac {dt}{t}}\right)^{\frac {1}{q}}}$ 

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

If ${\displaystyle p=q}$  and ${\displaystyle s}$  is not an integer, then ${\displaystyle B_{p,p}^{s}(\mathbf {R} )={\bar {W}}^{s,p}(\mathbf {R} )}$ , where ${\displaystyle {\bar {W}}^{s,p}(\mathbf {R} )}$  denotes the Sobolev–Slobodeckij space.

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