# 0.5 Approximation of functions

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We first give a definition of the order of a multiresolution analysis.

(Order of a MRA in the classical setting) A multiresolution analysis is said to be of order $\stackrel{˜}{N}$ if the primal scaling function $\varphi$ reproduces polynomials up to degree $\stackrel{˜}{N}-1$ , i.e., $\text{For}\phantom{\rule{1.em}{0ex}}0\le p<\stackrel{˜}{N},\phantom{\rule{0.277778em}{0ex}}\exists {c}_{k}\in \mathbb{R}\phantom{\rule{0.277778em}{0ex}}\text{such}\phantom{\rule{4.pt}{0ex}}\text{that}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{0.277778em}{0ex}}{x}^{p}=\sum _{k}{c}_{k}\varphi \left(x-k\right)\phantom{\rule{3.33333pt}{0ex}}.$

The associated dual wavelet $\stackrel{˜}{\psi }$ has then $\stackrel{˜}{N}$ vanishing moments.In the classical setting, it is proved that the order of a MRA and the regularity of the scaling function are linked: the larger $\stackrel{˜}{N}$ , the higher the regularity of $\varphi$ . Symmetrically to [link] , the order of the dual MRA is $N$ if $\stackrel{˜}{\varphi }$ can reproduce polynomials up to degree $N-1$ . Figure 2 from Multiresolution analysis and wavelets shows an example of a biorthogonal basis where $\stackrel{˜}{N}=3$ and $N=1$ . It illustrates the link between a high number of vanishing moments of thedual wavelet $\stackrel{˜}{\psi }$ and the regularity of the primal scaling function $\varphi$ .

The main objective when decomposing a function in a wavelet series is to create a sparse representation of the function, that is, to obtain a decomposition where only a few number of detail coefficients are large', while the majority of the coefficients are close to zero. By large', we mean that the absolute value of the detail coefficient is large.

Near a singularity, large detail coefficients at different levels will be needed to recover the discontinuity. However, between points of singularity, we can hope to have small detail coefficients, in particular if the analyzing wavelets ${\stackrel{˜}{\psi }}_{jk}$ have a large number $\stackrel{˜}{N}$ of vanishing moments. Indeed, suppose the function $f$ to be decomposed is analytic on the interval $I$ without discontinuity. Since $〈{x}^{p},\phantom{\rule{0.166667em}{0ex}},,,{\stackrel{˜}{\psi }}_{jk}〉=0$ for $p=0,...,\stackrel{˜}{N}-1$ , we are sure that the first $\stackrel{˜}{N}$ terms of a Taylor expansion of $f$ will not give a contribution to the wavelet coefficient $〈f,\phantom{\rule{0.166667em}{0ex}},,,{\stackrel{˜}{\psi }}_{jk}〉$ provided that the support of ${\stackrel{˜}{\psi }}_{jk}$ does not contain any singularities of the function $f$ .

This sparse representation explains why classical wavelets providesmoothness characterization of function spaces like the Hölder and Sobolev spaces [link] , but also of more general Besov spaces, which may contain functions of inhomogeneousregularity [link] , [link] , [link] , [link] , [link] .

We illustrate this characterization property with the case of $\beta -$ Hölder functions.

## Definition 2

The class ${\Lambda }^{\beta }\left(L\right)$ of Hölder continuous functions is defined as follows:
1. if $\beta \le 1,{\Lambda }^{\beta }\left(L\right)=\left\{f:\left|f,\left(,x,\right),-,f,\left(,y,\right)\right|\le L{|x-y|}^{\beta }\right\}$
2. if $\beta >1,{\Lambda }^{\beta }\left(L\right)=\left\{f,:,\left|{f}^{\left(⌊\beta ⌋\right)},\left(x\right),-,{f}^{\left(⌊\beta ⌋\right)},\left(y\right)\right|,\le ,{L|x-y|}^{{\beta }^{\text{'}}},\phantom{\rule{0.277778em}{0ex}},;,|{f}^{\left(⌊\beta ⌋\right)}\left(x\right)|,\le ,M\right\},$ where $⌊\beta ⌋$ is the largest integer less than $\beta$ and ${\beta }^{\text{'}}=\beta -⌊\beta ⌋.$

The global Hölder regularity of a function can be characterized as follows [link] , [link] .

Let $f\in {\Lambda }^{\beta }\left(L\right)$ , and suppose that the (orthogonal) wavelet function $\psi$ has $r$ continuous derivatives and $r$ vanishing moments with $r>\beta$ . Then
$\left|〈f,\phantom{\rule{0.166667em}{0ex}},,,{\psi }_{jk}〉\right|\le C{2}^{-j\left(\beta +1/2\right)}\phantom{\rule{3.33333pt}{0ex}}.$

A similar characterization exists for continuous and Sobolev functions [link] , [link] .

In the orthogonal setting, the wavelet $\psi$ must be regular and have a high number of vanishing moments. On the contrary,in the biorthogonal expansion equation 5 from Multiresolution analysis and wavelets , it is mostly of interest to have a dual wavelet $\stackrel{˜}{\psi }$ with a high number of vanishing moments, and hence a regular primal scaling and wavelet functions. On the primal side, it is sufficient to have only one vanishing moment for wavelet denoising, and consequently $\stackrel{˜}{\psi }$ may not be very regular. In this case, the wavelet coefficient $〈f,\phantom{\rule{0.166667em}{0ex}},,,{\stackrel{˜}{\psi }}_{jk}〉$ with the less regular wavelet ${\stackrel{˜}{\psi }}_{jk}$ can be used to characterize $f\in {\Lambda }^{\beta }\left(L\right)$ with $0<\beta <\stackrel{˜}{N}$ , even if $\beta >N=1$ : with a biorthogonal basis, regular functions can be characterized by their inner products with much less regular functions.

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