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An important feature of Besov spaces is that they admit equivalent characterization by multiresolution approximation properties and by wavelet decompositions.
Here we use the following standard notation (see [link] or [link] for a general treatment): if $f$ is function we denote by ${P}_{j}f$ its projection onto the space ${V}_{j}$ , and by ${Q}_{j}f={P}_{j+1}f-{P}_{j}f$ its projection onto the detail space ${W}_{j}$ . The multiscale decomposition of $f$ writes
The projectors ${P}_{j}$ and ${Q}_{j}$ can be further expressed in terms of biorthogonal scaling functions and wavelets bases:
Here we use the simplified notation ${\varphi}_{\lambda}$ with “ $\left|\lambda \right|=j$ ” meaning that the functions are picked at resolution $j$ . In the case where $\Omega ={\mathbb{R}}^{d}$ , thesehave the general from ${\varphi}_{\lambda}\left(x\right):={\varphi}_{j,k}\left(x\right):={2}^{dj/2}\varphi ({2}^{j}x-k)$ , bur for a general domain $\Omega ={\mathbb{R}}^{d}$ proper adaptations of these bases need to be done near the boundary. We can thereforewrite
where we include in this sum the wavelets at all levels $j\ge 0$ and we incorporate the scaling function ${\varphi}_{\lambda}$ at the first level $\left|\lambda \right|=0$ .
Under certain assumptions that we shall discuss below, it is known that the Besov norm ${\parallel f\parallel}_{{B}_{p,q}^{s}}$ is equivalent to
or to
Using the equivalence $\parallel {Q}_{j}{f\parallel}_{{L}^{p}}\sim {2}^{(d/2-d/p)j}{\parallel {\left({d}_{\lambda}\right)}_{\left|\lambda \right|=j}\parallel}_{{\ell}^{p}}$ at each level to prove a third equivalent norm interms of the wavelet coefficients:
These equivalences mean that the modulus of smoothness ${\omega}_{n}{(f,{2}^{-j})}_{{L}^{p}}$ in the definition of ${B}_{p,q}^{s}$ can be replaced either by $\parallel f-{P}_{j}{f\parallel}_{{L}^{p}}$ or by $\parallel {Q}_{j}{f\parallel}_{{L}^{p}}$ . Their validity requires thatthe spaces ${V}_{j}$ satisfy the following two assumptions:
One can show that the direct estimate is satisfied if and only if all polynomials up to order $n-1$ can be written as combinations of the scaling functions ${\varphi}_{\lambda}$ in ${V}_{j}$ , or equivalently if the dual wavelets ${\tilde{\psi}}_{\lambda}$ have $n$ vanishing moments. On the other hand, the inverse estimate requires that the scaling functions ${\varphi}_{\lambda}$ that generates ${V}_{j}$ are smooth in the sense of belonging to ${W}^{n,p}$ . Note that the direct estimate immediately implies that theexpression [link] is less than ${\parallel f\parallel}_{{B}_{p,q}^{s}}$ . A more refined mechanism, using theinverse estimate (as well as some discrete Hardy inequalities) is used to prove the full equivalence between ${\parallel f\parallel}_{{B}_{p,q}^{s}}$ and [link] or [link] . We refer to chapter III in [link] for a detailed proof of these results.
These equivalences show that the convergence rate ${N}^{-t/d}$ ( $N=\mathrm{dim}\left({V}_{j}\right)$ ) can be achieved by the linearmultiscale approximation process $f\mapsto {P}_{f}$ , if and only if the function has roughly “ $t$ derivatives in ${L}^{p}$ ”.
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