0.2 Besov spaces

The definition of “order of smoothness $s$ in $L^p$ ” for $s$ non-integer and $p$ different from 2 or $\infty$ is more subject to arbitrary choices. Among others, one may consider:

• Sobolev spaces $W^{s,p}$ defined (if $m<s<m+1$ ) by
  $${\parallel f\parallel }_{W^{s,p}}:={\left({\parallel f\parallel }_{W^{m,p}}^p,+,\sum _{\left|\alpha \right|=m},{\int }_{\Omega \times \Omega },\frac{|{\partial }^{\alpha }f\left(x\right)-{\partial }^{\alpha }{f\left(y\right)|}^p}{{|x-y|}^{(s-m)p+d}},d,x,d,y\right)}^{\frac{1}{p}}$$
  These spaces coincide with those defined by means of Fourier transform when $p=2$ (see [link] for a general treatment).
• Bessel-potential spaces $H^{s,p}$ defined by means of the Fourier transform operator $\mathcal{F}$ ,
  $${\parallel f\parallel }_{H^{s,p}}={\left({\parallel f\parallel }_{L^p}^p,+\parallel ,{\mathcal{F}}^{-1},{(1+|·|}^s,{)\mathcal{F}f\parallel }_{L^p}^p\right)}^{\frac{1p}{.}}$$
  These spaces coincides with the Sobolev spaces $W^{m,p}$ when $m$ is an integer and $1<p<+\infty$ (see [link] , p.38), but their definition requires that $\Omega ={\mathbb{R}}^d$ in order to apply the Fourier transform.
• Besov spaces $B_{p,q}^s$ , involving an extra parameter $q$ that we define below through finite differences. These spaces include most of thosethat we have listed so far as particular cases. As we shall see, one of their main interest is that they can be exactlycharacterized by multiresolution approximation error, as well as from the size properties of the wavelet coefficients.

We define the $n$ -th order $L^p$ modulus of smoothness of $f$ by

where ${\Omega }_{h,n}:=\{x\in \Omega ;x-kh\in \Omega ,k=0,\cdots ,n\}$ . Here we measure the “size” of ${\Delta }_h^{n}f$ in $L^p$ -norm, where we restrict to $L^p\left({\Omega }_{h,n}\right)$ to ensure that all the arguments $x-kh$ occurring in the computation of ${\Delta }_h^{n}f\left(x\right)$ still live in $\Omega$ . For $p,q\ge 1$ , $s>0$ , the Besov spaces $B_{p,q}^s$ consists of those functions $f\in L^p$ such that

Here $n$ is an integer strictly larger than $s$ . A natural norm for such a space is then given by

If $q=\infty$ , the condition [link] simply means that $\parallel {\Delta }_h^n{f\parallel }_{L^p}\le C{h}^{-s}$ for $\left|h\right|\le 1$ . For $q<\infty$ , the decay condition on ${\Delta }_h^{n}f$ is slightly stronger, since we require that the sequence ${\left(2^{sj}{\omega }_n{(f,2^{-j})}_{L^p}\right)}_{j\ge 0}^{q}i$ be summable. The space $B_{p,q}^s$ also represents “ $s$ order of smoothness measured in $L^p$ "; the parameter $q$ allows a finer tuning on the degree of smoothness - one has $B_{p,q_1}^s\subset B_{p,q_2}^s$ if $q_1\le q_2$ - but plays a minor role in comparison to $s$ since clearly

regardless of the values of $q_1$ and $q_2$ . Roughly speaking, smoothness of order $s$ in $L^p$ is expressed here by the fact that, for $n$ large enough, ${\omega }_n{(f,t)}_{L^p}$ goes to 0 like $\mathcal{O}\left(t^s\right)$ as $t\to 0$ .

Clearly ${\mathcal{C}}^s=B_{\infty ,\infty }^s$ when $s$ is not an integer. It can also be provedthat when $s$ is not an integer $W^{s,p}=B_{p,p}^s$ . These spaces are different from one another for integer values of $s$ , except when $p=2$ in which case $H^s=B_{2,2}^s$ for all values of $s$ (see [link] , p.38).