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The main interest of MRA lies in the fact that, whenever a collection of closed subspaces satisfies the conditions in [link] , there exists an orthonormal wavelet basis, noted $\{{\psi}_{j,k},k\in \mathrm{Z}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\mathrm{Z}\}$ of ${W}_{j}.$ Hence, we can say in very general terms that the projection of a function $f$ onto ${V}_{j+1}$ can be decomposed as
where ${\mathcal{P}}_{j}$ is the projection operator onto ${V}_{j},$ and ${\mathcal{Q}}_{j}$ is the projection operator onto ${W}_{j}.$ Before proceeding further, let us state clearly what the term “orthogonal wavelet ” involves.
We introduce above the class of orthogonal wavelets. Let us define this precisely.
An orthogonal wavelet basis is associated with an orthogonal multiresolution analysis that can be defined as follows. We talk about orthogonal MRA when the wavelet spaces ${W}_{j}$ are defined as the orthogonal complement of ${V}_{j}$ in ${V}_{j+1}.$ Consequently, the spaces ${W}_{j},$ with $j\in \mathrm{Z}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\mathrm{Z}$ are all mutually orthogonal, the projections ${\mathcal{P}}_{j}$ and ${\mathcal{Q}}_{j}$ are orthogonal, so that the expansion
is orthogonal. Moreover, as ${\mathcal{Q}}_{j}$ is orthogonal, the projection onto ${W}_{j}$ can explicitely be written as:
with
A sufficient condition for a MRA to be orthogonal is:
or $<\psi ,\varphi (.-l)>=0$ for $l\in \mathrm{Z}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\mathrm{Z},$ since the other conditions simply follow from scaling.
An orthogonal wavelet is a function $\psi $ such that the collection of functions $\left\{\psi \right(x-l\left)\right|l\in \mathrm{Z}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\mathrm{Z}\}$ is an orthonormal basis of ${W}_{0}.$ This is the case if $<\psi ,\psi (.-l)>={\delta}_{l,0}.$
In the following, we consider only orthogonal wavelets. We now outline how to construct a wavelet function $\psi \left(x\right)$ starting from $\varphi \left(x\right),$ and thereafter we show what this construction gives with the Haar function.
Suppose we have an orthonormal basis (ONB) $\{{\varphi}_{j,k},k\in \mathrm{Z}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\mathrm{Z}\}$ for ${V}_{j}$ and we want to construct ${\psi}_{jk}$ such that
It is natural to use conditions given by the MRA aspect to obtain this. More specifically, the following relationships are used to characterize $\psi :$
In fact, in our setting, the conditions given above can be re-written in the Fourier domain, where the manipulations become easier (for details,see [link] , chapter 5). Let us now state the result of these manipulations.
One possibility for the construction of the wavelet $\psi $ is to take
(with convergence of this serie is ${L}_{2}$ sense).
Let us see what the recipe of [link] gives for the Haar multiresolution analysis. In that case, $\varphi \left(x\right)=1$ for $0\le x\le 1,0$ otherwise. Hence:
Consequently,
If we consider a first (coarse) approximation of $f\in {V}_{0},$ and then “refine” this approximation with detail spaces ${W}_{j},$ the decomposition of $f$ can be written as:
where ${\alpha}_{k}=<f,{\varphi}_{0,k}>$ and ${\beta}_{j,k}=<f,{\psi}_{j,k}>.$ In this case, we talk about inhomogeneous representation of $f.$
If we use the fact that
we can decompose $f$ as a linear combination of functions ${\psi}_{j,k}$ only:
We then talk about homogeneous representation of $f$ .
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