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Besov space
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<h2 id="definition">Definition</h2> <p>Several equivalent definitions exist. One of them is given below. This definition is quite limited because it does not extend to the range <i>s</i> ≤ 0. </p><p>Let </p> Δ h f ( x ) = f ( x − h ) − f ( x ) {\displaystyle \Delta _{h}f(x)=f(x-h)-f(x)} <p>and define the <a href="/facts/Modulus_of_continuity/31lThXhW">modulus of continuity</a> by </p> ω p 2 ( f , t ) = sup | h | ≤ t ‖ Δ h 2 f ‖ p {\displaystyle \omega _{p}^{2}(f,t)=\sup _{|h|\leq t}\left\|\Delta _{h}^{2}f\right\|_{p}} <p>Let n be a non-negative integer and define: <i>s</i> = <i>n</i> + <i>α</i> with 0 < <i>α</i> ≤ 1. The Besov space B p , q s ( R ) {\displaystyle B_{p,q}^{s}(\mathbf {R} )} contains all functions f such that </p> f ∈ W n , p ( R ) , ∫ 0 ∞ | ω p 2 ( f ( n ) , t ) t α | q d t t < ∞ . {\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 .} <h2 id="norm">Norm</h2> <p>The Besov space B p , q s ( R ) {\displaystyle B_{p,q}^{s}(\mathbf {R} )} is equipped with the norm </p> ‖ f ‖ B p , q s ( R ) = ( ‖ f ‖ W n , p ( R ) q + ∫ 0 ∞ | ω p 2 ( f ( n ) , t ) t α | q d t t ) 1 q {\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}}} <p>The Besov spaces B 2 , 2 s ( R ) {\displaystyle B_{2,2}^{s}(\mathbf {R} )} coincide with the more classical <a href="/facts/Sobolev_spaces/34qxWCyr">Sobolev spaces</a> H s ( R ) {\displaystyle H^{s}(\mathbf {R} )} . </p><p>If p = q {\displaystyle p=q} and s {\displaystyle s} is not an integer, then B p , p s ( R ) = W ¯ s , p ( R ) {\displaystyle B_{p,p}^{s}(\mathbf {R} )={\bar {W}}^{s,p}(\mathbf {R} )} , where W ¯ s , p ( R ) {\displaystyle {\bar {W}}^{s,p}(\mathbf {R} )} denotes the <a href="/facts/Sobolev_space/34qxWCyr">Sobolev–Slobodeckij space</a>. </p> <ul><li>Triebel, Hans (1992). <i>Theory of Function Spaces II</i>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-0346-0419-2">10.1007/978-3-0346-0419-2</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-0346-0418-5.</li> <li>Besov, O. V. (1959). "On some families of functional spaces. Imbedding and extension theorems". <i>Dokl. Akad. Nauk SSSR</i> (in Russian). 126: 1163–1165. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0107165">0107165</a>.</li> <li>DeVore, R. and Lorentz, G. "Constructive Approximation", 1993.</li> <li>DeVore, R., Kyriazis, G. and Wang, P. "Multiscale characterizations of Besov spaces on bounded domains", Journal of Approximation Theory 93, 273-292 (1998).</li> <li>Leoni, Giovanni (2017). <i><a href="http://bookstore.ams.org/gsm-181/">A First Course in Sobolev Spaces: Second Edition</a></i>. <a href="/facts/Graduate_Studies_in_Mathematics/APstozeU">Graduate Studies in Mathematics</a>. 181. American Mathematical Society. pp. 734. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4704-2921-8</li></ul>