0.3 Approximation of functions with wavelets

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Approximation of functions with wavelets

In Example from Multiresolution analysis , we saw the Haar wavelet basis. This is the simplest wavelet one can imagine, but its approximation properties are not very good. Indeed, the accuracy of the approximation is somehow related to the regularity of the functions $\psi$ and $\varphi .$ We will show this for different settings: in case the function $f$ is continuous, belongs to a Sobolev or to a Hölder space. But first we introduce the notion of regularity of a MRA.

Regularity of a multiresolution analysis

For wavelet bases (orthonormal or not), there is a link between the regularity of $\psi$ and the number of vanishing moments. More precisely, we have the following:

Let $\left\{{\psi }_{j,k}\right\}$ be an orthonormal basis (ONB) in ${L}^{2}\left(\mathrm{I}\phantom{\rule{-0.166667em}{0ex}}\mathrm{R}\right),$ with $\psi \in C\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\left[0.1ex\right]0.05em1.25ex\phantom{\rule{0.277778em}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}^{m},{\psi }^{\left(l\right)}$ bounded for $l\le m$ and $|\psi \left(x\right)|\le C{\left(1+|x|\right)}^{-\alpha }$ for $\alpha >m+1.$ Then we have:
$\int {x}^{l}\psi \left(x\right)dx=0\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}l=0,1,...,m.$

(this describes the “decay in frequency domain”).

This proposition implies the following corollary:

Suppose the $\left\{{\psi }_{j,k}\right\}$ are orthonormal. Then it is impossible that $\psi$ has exponential decay, and that $\psi \in C\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\left[0.1ex\right]0.05em1.25ex\phantom{\rule{0.277778em}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}^{\infty },$ with all the derivatives bounded, unless $\psi \equiv 0$

This corollary tells us that a trade-off has to be done: we have to choose for exponential (or faster) decay in, either time or frequency domain; we cannot have both. We now come to the definition of a $r-$ regular MRA (see [link] ).

(Meyer, 1990) A MRA is called r-regular ( $r\in \mathrm{I}\phantom{\rule{-0.166667em}{0ex}}\mathrm{N}$ ) when $\varphi \in C\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\left[0.1ex\right]0.05em1.25ex\phantom{\rule{0.277778em}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}^{r}$ and for all $m\in \mathrm{I}\phantom{\rule{-0.166667em}{0ex}}\mathrm{N},$ there exists a constant ${C}_{m}>0$ such that:
$|\frac{{\partial }^{\alpha }\varphi \left(x\right)}{\partial {x}^{\alpha }}|\le {C}_{m}{\left(1+|x|\right)}^{-m}\forall \alpha \le r$

If one has a $r-$ regular MRA, then the corresponding wavelet $\psi \left(x\right)\in C\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\left[0.1ex\right]0.05em1.25ex\phantom{\rule{0.277778em}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}^{r},$ satisfies [link] and has $r$ vanishing moments:

$\int {x}^{k}\psi \left(x\right)dx=0,k=0,1,...,r.$

We now have the tools needed to measure the decay of approximation error when the resolution (or the finest level) increases.

Approximation of a continuous function

Let $\left\{{\psi }_{j,k}\right\}$ come from a r-regular MRA. Then, we have, for a continuous function $f\in C\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\left[0.1ex\right]0.05em1.25ex\phantom{\rule{0.277778em}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}^{s}\left(0 the following:
$||=O\left({2}^{-j\left(s+1/2\right)}\right)$
( $s=1$ ). As $\int \overline{\psi \left(x\right)}=0$ and $|f\left(x\right)-f\left(k/{2}^{j}\right)|\le C|x-k/{2}^{j}|,$ ( $f$ is Lipschitz continuous), we have:
$\begin{array}{ccc}\hfill ||& =& |{2}^{j/2}\int \left[f\left(x\right)-f\left(k/{2}^{j}\right)\right]\overline{\psi }\left({2}^{j}x-k\right)dx|,k\in \mathrm{Z}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\mathrm{Z}\hfill \\ & \le & C{2}^{j/2}\int |x-k/{2}^{j}||\overline{\psi }\left({2}^{j}x-k\right)|dx\hfill \\ & =& C{2}^{-j/2}\int |{2}^{j}x-k||\overline{\psi }\left({2}^{j}x-k\right)|dx\hfill \\ & & \left(\phantom{\rule{4.pt}{0ex}}\text{let}\phantom{\rule{4.pt}{0ex}}u={2}^{j}x-k\right)\hfill \\ & =& {2}^{-j\left(1+1/2\right)}C\int |u||\overline{\psi }\left(u\right)|du<\infty \hfill \\ & =& O\left({2}^{-j\left(1+1/2\right)}\right).\hfill \end{array}$

For $s>1,$ we use proposition [link] and we iterate. $\square$

Note that the reverse of [link] is true: [link] entails that $f$ is in ${C}^{s}.$

Under the assumptions of [link] , the error of approximation of a function $f\in C\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\left[0.1ex\right]0.05em1.25ex\phantom{\rule{0.277778em}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}^{s}$ at scale ${V}_{J}$ is given by:
$||f-\sum _{k}{\varphi }_{J,k}{||}_{2}=O\left({2}^{-Js}\right)$
Using the exact decomposition of $f$ given by Equation 30 from Multiresolution Analysis , one has:
$\begin{array}{ccc}\hfill ||f-{\mathcal{P}}_{J}f{||}_{2}& =& ||\sum _{j\ge J}\sum _{k}{\psi }_{j,k}{||}_{2}\hfill \\ & \le & \sum _{j\ge J}C{2}^{-j\left(s+1/2\right)}||\sum _{k}{\psi }_{j,k}{||}_{2}\hfill \end{array}$

Now, if we suppose that $|\psi \left(k\right)|\le {C}^{\text{'}}{\left(1+|k|\right)}^{-m},\left(m\ge s+1\right),$ we obtain:

$||\sum _{k}{\psi }_{j,k}{||}_{2}\le {C}^{\text{'}}{2}^{j/2}$

$||f-{\mathcal{P}}_{J}f{||}_{2}\le {C}^{\text{'}\text{'}}\sum _{j\ge J}{2}^{-js}={C}^{\text{'}\text{'}}{\left(1-{2}^{-s}\right)}^{-1}{2}^{-Js}=O\left({2}^{-Js}\right)\square$

Hence, we verify with this last corollary that, as $J$ increases, the approximation of $f$ becomes more accurate.

Approximation of functions in sobolev spaces

Let us first recall the definition of weak differentiability, for this notion intervenes in the definition of a Sobolev space.

Let $f$ be a function defined on the real line which is integrable on every bounded interval. If there exists a function $g$ defined on the real line which is integrable on every bounded interval such that:
$\forall x\le y,{\int }_{x}^{y}g\left(u\right)du=f\left(y\right)-f\left(x\right),$

then the function $f$ is called weakly differentiable. The function $g$ is defined almost everywhere, is called the weak derivative of $f$ and will be denoted by ${f}^{\text{'}}$ .

A function $f$ is $N$ -times weakly differentiable if it has derivatives $f,{f}^{\text{'}},...,{f}^{\left(N-1\right)}$ which are continuous and ${f}^{\left(N\right)}$ which is a weak derivative.

We are now able to define Sobolev spaces.

Let $1\le p<\infty ,m\in \left\{0,1,...\right\}.$ The function $f\in {L}_{p}\left(\mathrm{I}\phantom{\rule{-0.166667em}{0ex}}\mathrm{R}\right)$ belongs to the Sobolev space ${W}_{p}^{m}\left(\mathrm{I}\phantom{\rule{-0.166667em}{0ex}}\mathrm{R}\right)$ if it is m-times weakly differentiable, and if ${f}^{\left(m\right)}\in {L}_{p}\left(\mathrm{I}\phantom{\rule{-0.166667em}{0ex}}\mathrm{R}\right).$ In particular, ${W}_{p}^{0}\left(\mathrm{I}\phantom{\rule{-0.166667em}{0ex}}\mathrm{R}\right)={L}_{p}\left(\mathrm{I}\phantom{\rule{-0.166667em}{0ex}}\mathrm{R}\right).$

The approximation properties of wavelet expansions on Sobolev spaces are given, among other, in Härdle et.al (see [link] ). Suppose we have at our disposal a scaling function $\varphi$ which generates a MRA. The approximation theorem can be stated as follows:

(Approx. in Sobolev space) Let $\varphi$ be a scaling function such that $\left\{\varphi \left(.-k\right),k\in \mathrm{Z}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\mathrm{Z}\right\}$ is an ONB and the corresponding spaces ${V}_{j}$ are nested. In addition, let $\varphi$ be such that
${\int |\varphi \left(x\right)||x|}^{N+1}dx<\infty ,$

and let at least one of the following assumptions hold:

1. $\varphi \in {W}_{q}^{N}\left(\mathrm{I}\phantom{\rule{-0.166667em}{0ex}}\mathrm{R}\right)\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{some}\phantom{\rule{4.pt}{0ex}}1\le q<\infty$
2. $\int {x}^{n}\psi \left(x\right)dx=0,n=0,1,...,N,$ where $\psi$ is the mother wavelet associated to $\varphi .$

Then, if $f$ belongs to the Sobolev space ${W}_{p}^{N+1}\left(\mathrm{I}\phantom{\rule{-0.166667em}{0ex}}\mathrm{R}\right),$ we have:

$||{\mathcal{P}}_{j}f-f{||}_{p}=O\left({2}^{-j\left(N+1\right)}\right),\phantom{\rule{4.pt}{0ex}}\text{as}\phantom{\rule{4.pt}{0ex}}j\to \infty ,$

where ${\mathcal{P}}_{j}$ is the projection operator onto ${V}_{j}.$

Approximation of functions in hölder spaces

Here we assume for simplicity that $\psi$ has compact support and is $C\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\left[0.1ex\right]0.05em1.25ex\phantom{\rule{0.277778em}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}^{1}$ (the formulation of the theorems are slightly different for more general $\psi$ ).

If $f$ is Hölder continuous with exponent $\alpha ,0<\alpha <1$ at ${x}_{0},$ then
$\underset{k}{max}\left\{||dist{\left({x}_{0},supp\left({\psi }_{j,k}\right)\right)}^{-\alpha }\right\}=O\left({2}^{-j\left(1/2+\alpha \right)}\right)$

The reverse of theorem [link] does not exactly hold: we must modify condition [link] slightly. More precisely, we have the following:

Define, for $ϵ>0$ , the set
$S\left({x}_{0},j,ϵ\right)=\left\{k\in \mathrm{Z}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\mathrm{Z}|supp\left({\psi }_{j,k}\right)\cap \right]{x}_{0}-ϵ,{x}_{0}+ϵ\left[\ne \varnothing \right\}.$

If, for some $ϵ>0$ and some $\alpha \left(0<\alpha <1\right),$

$\underset{k\in S\left({x}_{0},j,ϵ\right)}{max}||=O\left({2}^{-j\left(1/2+\alpha \right)}\right),$

then $f$ is Hölder continuous with exponent $\alpha$ at ${x}_{0}.$

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