Besov's space

Spaces of Besov $B_{p,q}^{s}(\mathbb {R} )$ ${\ displaystyle B_ {p, q} ^ {s} (\ mathbb {R})}$ ${\ displaystyle B_ {p, q} ^ {s} (\ mathbb {R})}$ - complete spaces of functions that are Banach for 1 ≤ p , q ≤ ∞ . Named in honor of the developer - Soviet mathematician Oleg Vladimirovich Besov . These spaces, along with the Tribel – Lizorkin spaces defined in a similar way, are a generalization of simpler function spaces and are used to determine the regularity of functions.

Definition

There are several equivalent definitions, one of them is given here.

Let be

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

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

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

and the modulus of continuity is defined as

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

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

\omega^2_p(f,t) = \sup_{|h| \le t} \left \| \Delta^2_h f \right \|_p.

Let n be a non-negative integer and s = n + α with 0 < α ≤ 1 . Besov's space $B_{p,q}^{s}(\mathbb {R} )$ ${\ displaystyle B_ {p, q} ^ {s} (\ mathbb {R})}$ $B_{p,q}^{s}(\mathbb {R} )$ consists of functions f such that

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

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

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

Where ${\bar {W}}^{n,p}(\mathbb {R} )$ ${\ displaystyle {\ bar {W}} ^ {n, p} (\ mathbb {R})}$ ${\bar {W}}^{n,p}(\mathbb {R} )$ - Sobolev space .

Norma

In the Besov space $B_{p,q}^{s}(\mathbb {R} )$ ${\ displaystyle B_ {p, q} ^ {s} (\ mathbb {R})}$ $B_{p,q}^{s}(\mathbb {R} )$ there is a norm

\left\|f\right\|_{B_{p,q}^{s}(\mathbb {R} )}=\left(\|f\|_{W^{n,p}(\mathbb {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}}

{\ displaystyle \ left \ | f \ right \ | _ {B_ {p, q} ^ {s} (\ mathbb {R})} = \ left (\ | f \ | _ {W ^ {n, p} (\ mathbb {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}}}

\left\|f\right\|_{B_{p,q}^{s}(\mathbb {R} )}=\left(\|f\|_{W^{n,p}(\mathbb {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}}

Spaces of Besov $B_{2,2}^{s}(\mathbb {R} )$ ${\ displaystyle B_ {2,2} ^ {s} (\ mathbb {R})}$ $B_{2,2}^{s}(\mathbb {R} )$ coincide with more ordinary Sobolev spaces $H^{s}(\mathbb {R} )$ ${\ displaystyle H ^ {s} (\ mathbb {R})}$ $H^{s}(\mathbb {R} )$ .

If a $p=q$ ${\ displaystyle p = q}$ $p=q$ and $s$ ${\ displaystyle s}$ $s$ Is not an integer then $B_{p,p}^{s}(\mathbb {R} )={\bar {W}}^{s,p}(\mathbb {R} )$ ${\ displaystyle B_ {p, p} ^ {s} (\ mathbb {R}) = {\ bar {W}} ^ {s, p} (\ mathbb {R})}$ $B_{p,p}^{s}(\mathbb {R} )={\bar {W}}^{s,p}(\mathbb {R} )$ where ${\bar {W}}^{s,p}(\mathbb {R} )$ ${\ displaystyle {\ bar {W}} ^ {s, p} (\ mathbb {R})}$ ${\bar {W}}^{s,p}(\mathbb {R} )$ - Sobolev space .

Embedding Theorem

Let be ${\displaystyle 1<semantics><mrow class=$ ${\ displaystyle 1 <p \ leqslant q <\ infty}$ , $-\infty <t\leqslant s<\infty$ ${\ displaystyle - \ infty <t \ leqslant s <\ infty}$ , $r\in [1,\infty ]$ ${\ displaystyle r \ in [1, \ infty]}$ .

If equality holds $s-n/p=t-n/q,$ ${\ displaystyle sn / p = tn / q,}$ then there is continuous investment

$B_{p,r}^{s}(\mathbb {R} ^{n})\subset B_{q,r}^{t}(\mathbb {R} ^{n}).$ ${\ displaystyle B_ {p, r} ^ {s} (\ mathbb {R} ^ {n}) \ subset B_ {q, r} ^ {t} (\ mathbb {R} ^ {n}).}$

If a $s=n/p+t$ ${\ displaystyle s = n / p + t}$ , $t>0$ ${\ displaystyle t> 0}$ and at least one of two conditions is satisfied: $r=1$ ${\ displaystyle r = 1}$ or $t$ ${\ displaystyle t}$ not an integer, then the embedding is true

$B_{p,r}^{s}(\mathbb {R} ^{n})\subset C^{t}(\mathbb {R} ^{n}).$ ${\ displaystyle B_ {p, r} ^ {s} (\ mathbb {R} ^ {n}) \ subset C ^ {t} (\ mathbb {R} ^ {n}).}$

Note : when $s<0,\ q\neq 1$ ${\ displaystyle s <0, \ q \ neq 1}$ space $B_{p,r}^{s}(\mathbb {R} ^{n})$ ${\ displaystyle B_ {p, r} ^ {s} (\ mathbb {R} ^ {n})}$ can be understood as a space conjugate to $B_{p',r'}^{s'}(\mathbb {R} ^{n})$ ${\ displaystyle B_ {p ', r'} ^ {s'} (\ mathbb {R} ^ {n})}$ where $s'=-s,\ 1/p+1/p'=1,\ 1/r+1/r'=1.$ ${\ displaystyle s' = - s, \ 1 / p + 1 / p '= 1, \ 1 / r + 1 / r' = 1.}$

Interpolation of Besov spaces

Let be $s_{0},s_{1}\in \mathbb {R}$ ${\ displaystyle s_ {0}, s_ {1} \ in \ mathbb {R}}$ , $s_{0}\neq s_{1},\ p\in (1,\infty )$ ${\ displaystyle s_ {0} \ neq s_ {1}, \ p \ in (1, \ infty)}$ , $q_{1},q_{2},q\in [1,\infty ],\ \theta \in (0,1)$ ${\ displaystyle q_ {1}, q_ {2}, q \ in [1, \ infty], \ \ theta \ in (0,1)}$ .

Then for interpolation spaces the following equality is true $\left(B_{p,q_{1}}^{s_{0}}(\mathbb {R} ^{n}),B_{p,q_{2}}^{s_{1}}(\mathbb {R} ^{n})\right)_{\theta ,q}=B_{p,q}^{(1-\theta )s_{0}+\theta s_{1}}(\mathbb {R} ^{n}).$ ${\ displaystyle \ left (B_ {p, q_ {1}} ^ {s_ {0}} (\ mathbb {R} ^ {n}), B_ {p, q_ {2}} ^ {s_ {1}} (\ mathbb {R} ^ {n}) \ right) _ {\ theta, q} = B_ {p, q} ^ {(1- \ theta) s_ {0} + \ theta s_ {1}} (\ mathbb {R} ^ {n}).}$

Literature

O.V. Besov, “On a certain family of function spaces. Embedding and Continuation Theorems ”, Dokl. USSR Academy of Sciences, 126: 6 (1959), 1163–1165
Triebel, H. "Theory of Function Spaces II". (eng.)
Tribel, H. "Interpolation Theory, Function Spaces, Differential Operators", 1980.
DeVore, R. and Lorentz, G. "Constructive Approximation", 1993. (English)
DeVore, R., Kyriazis, G. and Wang, P. " Multiscale characterizations of Besov spaces on bounded domains ", Journal of Approximation Theory 93, 273-292 (1998). (eng.)

Links

9.2 Besov Spaces / D. Picard, Wavelets, Approximation, and Statistical Applications (translation by K. A. Alekseev), ISBN 978-1-4612-2222-4 , 1998