# 0.4 Characterization by approximation properties

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The following is a short introduction to Besov spaces and their characterization by means of approximationprocedures as well as wavelet decompositions.

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

$f={P}_{0}f+\sum _{j\ge 0}{Q}_{j}f.$

The projectors ${P}_{j}$ and ${Q}_{j}$ can be further expressed in terms of biorthogonal scaling functions and wavelets bases:

${P}_{j}f:=\sum _{|\lambda |=j}⟨f,{\stackrel{˜}{\varphi }}_{\lambda }⟩{\varphi }_{\lambda }\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{and}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{Q}_{j}f:=\sum _{|\lambda |=j}⟨f,{\stackrel{˜}{\psi }}_{\lambda }⟩{\psi }_{\lambda }.$

Here we use the simplified notation ${\varphi }_{\lambda }$ with “ $|\lambda |=j$ ” meaning that the functions are picked at resolution $j$ . In the case where $\Omega ={\mathbb{R}}^{d}$ , thesehave the general from ${\varphi }_{\lambda }\left(x\right):={\varphi }_{j,k}\left(x\right):={2}^{dj/2}\varphi \left({2}^{j}x-k\right)$ , bur for a general domain $\Omega ={\mathbb{R}}^{d}$ proper adaptations of these bases need to be done near the boundary. We can thereforewrite

$f=\sum {d}_{\lambda }{\psi }_{\lambda },\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{d}_{\lambda }:=⟨f,{\stackrel{˜}{\psi }}_{\lambda }⟩,$

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 $|\lambda |=0$ .

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

$\parallel {P}_{0}{f\parallel }_{{L}^{p}}+\parallel \left({2}^{sj}\parallel f-{P}_{j}{f\parallel }_{{L}^{p}}{{\right)}_{j\ge 0}\parallel }_{{\ell }^{q}},$

or to

$\parallel {P}_{0}{f\parallel }_{{L}^{p}}+\parallel \left({2}^{sj}\parallel {Q}_{j}{f\parallel }_{{L}^{p}}{{\right)}_{j\ge 0}\parallel }_{{\ell }^{q}}.$

Using the equivalence $\parallel {Q}_{j}{f\parallel }_{{L}^{p}}\sim {2}^{\left(d/2-d/p\right)j}{\parallel {\left({d}_{\lambda }\right)}_{|\lambda |=j}\parallel }_{{\ell }^{p}}$ at each level to prove a third equivalent norm interms of the wavelet coefficients:

$\parallel \left({2}^{sj}{2}^{\left(d/2-d/p\right)j}\parallel {\left({d}_{\lambda }\right)}_{|\lambda |=j}{\parallel }_{{\ell }^{p}}{{\right)}_{j\ge 0}\parallel }_{{\ell }^{q}}.$

These equivalences mean that the modulus of smoothness ${\omega }_{n}{\left(f,{2}^{-j}\right)}_{{L}^{p}}$ 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:

• The ${V}_{j}$ must satisfy an approximation property that takes the form of a direct estimate
$\parallel f-{P}_{j}{f\parallel }_{{L}^{p}}\le C{\omega }_{n}{\left(f,{2}^{-j}\right)}_{{L}^{p}}.$
Such an estimate ensures that a smooth function will have a fast rate of approximation.
• They must also satisfy smoothness properties that takes the form of an inverse estimate
${\omega }_{n}{\left({f}_{j},t\right)}_{{L}^{p}}\le C{\left[min\left(1,t{2}^{j}\right)\right]}^{n}{\parallel {f}_{j}\parallel }_{{L}^{p}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{if}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{f}_{j}\in {V}_{j}.$
Such an estimate takes into account the smoothness of the spaces ${V}_{j}$ : it ensuresthat a function that is approximated at a sufficiently fast rate rate by these spacesshould also have some smoothness.

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 ${\stackrel{˜}{\psi }}_{\lambda }$ 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 $f↦{P}_{f}$ , if and only if the function has roughly “ $t$ derivatives in ${L}^{p}$ ”.

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