0.3 Approximation of functions with wavelets

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:

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

This proposition implies the following corollary:

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] ).

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

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

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 $C^s.$

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

Putting inequalities [link] and [link] together, we have:

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.

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 $f^{\text{'}}$ .

We are now able to define Sobolev spaces.

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:

and let at least one of the following assumptions hold:

1. $\varphi \in W_q^N\left(\mathrm{I}\mathrm{R}\right)\text{for}\text{some}1\le q<\infty$
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 $W_p^{N+1}\left(\mathrm{I}\mathrm{R}\right),$ we have:

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 $C[0.1ex]0.05em1.25ex{}^1$ (the formulation of the theorems are slightly different for more general $\psi$ ).

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

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

then $f$ is Hölder continuous with exponent $\alpha$ at $x_0.$