Besov room

A Besov room (after Oleg Wladimirowitsch Bessow ) is a function room . Like the similarly defined Lizorkin-Triebel space, it serves to define generalized function spaces by measuring (in a certain way) smoothness properties of the functions. The spectrogram is clearly subdivided into exponentially larger sections, the size of which is in turn determined on the basis of their spectrograms. ${\displaystyle B_{p,q}^{s}(\mathbb {R} ^{n})}$

preparation

It is so one exists partition of unity on the properties ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle (\varphi _{i})_{i\in \mathbb {N} }\subset C_{0}^{\infty }(\mathbb {R} ^{n})}$ ${\displaystyle \mathbb {R} ^{n}}$

• ${\displaystyle \operatorname {supp} (\varphi _{0})\subset B_{2}(0)}$,
• ${\displaystyle \operatorname {supp} (\varphi _{j})\subset \{\xi \in \mathbb {R} ^{n}:2^{j-1}\leq \left|\xi \right|\leq 2^{j+1}\}}$for all , ${\displaystyle j\geq 1}$
• ${\displaystyle \varphi _{j}(\xi )=\varphi _{1}\left({\frac {\xi }{2^{j-1}}}\right)}$.

Be the Schwartz room . For we define ${\displaystyle {\mathcal {S}}(\mathbb {R} ^{n})}$ ${\displaystyle f\in {\mathcal {S}}(\mathbb {R} ^{n})}$

${\displaystyle \varphi _{j}(D_{x})f:={\mathcal {F}}^{-1}\left[\varphi _{j}(\xi ){\mathcal {F}}\left[f\right]\right]}$for all , ${\displaystyle j\geq 0}$

where and denote the Fourier transform or its inverse. For functions from the dual space we define ${\displaystyle {\mathcal {F}}}$ ${\displaystyle {\mathcal {F}}^{-1}}$ ${\displaystyle f\in {\mathcal {S}}'(\mathbb {R} ^{n})}$

${\displaystyle \left\langle \varphi _{j}(D_{x})f,\psi \right\rangle _{{\mathcal {S}}'(\mathbb {R} ^{n}),{\mathcal {S}}(\mathbb {R} ^{n})}:=\left\langle f,\varphi _{j}(D_{x})\psi \right\rangle _{{\mathcal {S}}'(\mathbb {R} ^{n}),{\mathcal {S}}(\mathbb {R} ^{n})}}$for everyone and for everyone . ${\displaystyle j\geq 0}$ ${\displaystyle \psi \in {\mathcal {S}}(\mathbb {R} ^{n})}$

According to Paley-Wiener theorem , there is a function, since its Fourier transform has a compact support. ${\displaystyle \varphi _{j}(D_{x})f}$ ${\displaystyle C^{\infty }(\mathbb {R} ^{n})}$

definition

Be , and . Then we define ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle s\in \mathbb {R} }$ ${\displaystyle 1\leq p,q\leq \infty }$

${\displaystyle B_{pq}^{s}(\mathbb {R} ^{n}):=\left\{f\in {\mathcal {S}}'(\mathbb {R} ^{n}):\left\Vert f\right\Vert _{B_{pq}^{s}(\mathbb {R} ^{n})}<\infty \right\}}$,

where denotes the dual space of the Schwartz functions and ${\displaystyle {\mathcal {S}}'(\mathbb {R} ^{n})}$

${\displaystyle \left\Vert f\right\Vert _{B_{pq}^{s}(\mathbb {R} ^{n})}:=\left\Vert 2^{js}\left\Vert \varphi _{j}(D_{x})f\right\Vert _{L^{p}(\mathbb {R} ^{n})}\right\Vert _{l^{q}(\mathbb {N} )}}$.

properties

Besov spaces are (generally not separable) Banach spaces. Be , then applies ${\displaystyle s\in \mathbb {R} }$

${\displaystyle B_{2,2}^{s}(\mathbb {R} ^{n})=H_{2}^{s}(\mathbb {R} ^{n})}$.

Thus, the Besov spaces defined above are indeed a generalization of the classical Lebesgue spaces and Sobolev spaces . Furthermore applies to ${\displaystyle 0<s<1}$

${\displaystyle B_{\infty ,\infty }^{s}(\mathbb {R} ^{n})=C^{s}(\mathbb {R} ^{n})}$.

The equivalence applies to with ${\displaystyle r,s\in \mathbb {R} }$

1. Young's condition applies ${\displaystyle r+s>0}$
2. The multiplication mapping can clearly be continued to a continuous bilinear mapping . ${\displaystyle {\mathcal {S}}(\mathbb {R} ^{n})\times {\mathcal {S}}(\mathbb {R} ^{n})\longrightarrow {\mathcal {S}}(\mathbb {R} ^{n}),(\phi ,\psi )\longmapsto \phi \psi }$ ${\displaystyle C^{r}(\mathbb {R} ^{n})\times C^{s}(\mathbb {R} ^{n})\longrightarrow C^{r\wedge s}(\mathbb {R} ^{n})}$

Embeddings

Be , and . Then applies ${\displaystyle s\in \mathbb {R} }$ ${\displaystyle 1\leq p,q_{0},q_{1}\leq \infty }$ ${\displaystyle \varepsilon >0}$

• ${\displaystyle B_{p,q_{0}}^{s}(\mathbb {R} ^{n})\hookrightarrow B_{p,q_{1}}^{s}(\mathbb {R} ^{n})}$for , ${\displaystyle q_{0}\leq q_{1}}$
• ${\displaystyle B_{p,\infty }^{s+\varepsilon }(\mathbb {R} ^{n})\hookrightarrow B_{p,1}^{s}(\mathbb {R} ^{n})}$.

For , applies ${\displaystyle s\in \mathbb {R} }$ ${\displaystyle 1<p,q<\infty }$

• ${\displaystyle B_{p,p}^{s}(\mathbb {R} ^{n})\hookrightarrow H_{p}^{s}(\mathbb {R} ^{n})\hookrightarrow B_{p,2}^{s}(\mathbb {R} ^{n})}$for , ${\displaystyle 1<p\leq 2}$
• ${\displaystyle B_{p,2}^{s}(\mathbb {R} ^{n})\hookrightarrow H_{p}^{s}(\mathbb {R} ^{n})\hookrightarrow B_{p,p}^{s}(\mathbb {R} ^{n})}$for . ${\displaystyle 2\leq p<\infty }$

swell

• Triebel, H . "Theory of Function Spaces II"; ISBN 978-0817626396 .
• Besov, OV "On a certain family of functional spaces. Embedding and extension theorems", Docl. Akad. Nauk SSSR 126 (1959), 1163-1165.
• DeVore, R. and Lorentz, G. "Constructive Approximation", 1993; ISBN 978-3540506270 .
• DeVore, R., Kyriazis, G. and Wang, P. "Multiscale characterizations of Besov spaces on bounded domains", Journal of Approximation Theory 93, 273-292 (1998).