Editing Wavelet (section)

=== Multiresolution based discrete wavelet transforms (continuous in time)===
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I'm not sure content matches section header... weirdness.
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[[Image:Daubechies4-functions.svg|thumb|right|D4 wavelet]]
In any discretised wavelet transform, there are only a finite number of wavelet coefficients for each bounded rectangular region in the upper halfplane. Still, each coefficient requires the evaluation of an integral. In special situations this numerical complexity can be avoided if the scaled and shifted wavelets form a [[multiresolution analysis]]. This means that there has to exist an [[auxiliary function]], the ''father wavelet'' φ in ''L''<sup>2</sup>('''R'''), and that ''a'' is an integer. A typical choice is ''a'' = 2 and ''b'' = 1. The most famous pair of father and mother wavelets is the [[Daubechies wavelets|Daubechies]] 4-tap wavelet. Note that not every orthonormal discrete wavelet basis can be associated to a multiresolution analysis; for example, the Journe wavelet admits no multiresolution analysis.<ref>{{citation |last1=Larson |first1=David R. |year=2007 |title= Wavelet Analysis and Applications (See: Unitary systems and wavelet sets) |series = Appl. Numer. Harmon. Anal. |publisher=Birkhäuser |pages=143–171 }}</ref>

From the mother and father wavelets one constructs the subspaces
<math display="block">V_m=\operatorname{span}(\phi_{m,n}:n\in\Z),\text{ where }\phi_{m,n}(t)=2^{-m/2}\phi(2^{-m}t-n)</math>
<math display="block">W_m=\operatorname{span}(\psi_{m,n}:n\in\Z),\text{ where }\psi_{m,n}(t)=2^{-m/2}\psi(2^{-m}t-n).</math>
The father wavelet <math>V_{i}</math> keeps the time domain properties, while the mother wavelets <math>W_{i}</math> keeps the frequency domain properties.

From these it is required that the sequence
<math display="block">\{0\}\subset\dots\subset V_{1}\subset V_{0}\subset V_{-1}\subset V_{-2}\subset\dots\subset L^2(\R)</math>
forms a [[multiresolution analysis]] of ''L<sup>2</sup>'' and that the subspaces <math>\dots,W_1,W_0,W_{-1},\dots</math> are the orthogonal "differences" of the above sequence, that is, ''W<sub>m</sub>'' is the orthogonal complement of ''V<sub>m</sub>'' inside the subspace ''V''<sub>''m''−1</sub>,
<math display="block">V_m\oplus W_m=V_{m-1}.</math>

In analogy to the [[sampling theorem]] one may conclude that the space ''V<sub>m</sub>'' with sampling distance 2<sup>''m''</sup> more or less covers the frequency baseband from 0 to 1/2<sup>''m''-1</sup>. As orthogonal complement, ''W<sub>m</sub>'' roughly covers the band [1/2<sup>''m''−1</sup>, 1/2<sup>''m''</sup>].

From those inclusions and orthogonality relations, especially <math>V_0\oplus W_0=V_{-1}</math>, follows the existence of sequences <math>h=\{h_n\}_{n\in\Z}</math> and <math>g=\{g_n\}_{n\in\Z}</math> that satisfy the identities
<math display="block">g_n=\langle\phi_{0,0},\,\phi_{-1,n}\rangle</math> so that <math display="inline">\phi(t)=\sqrt2 \sum_{n\in\Z} g_n\phi(2t-n),</math> and
<math display="block">h_n=\langle\psi_{0,0},\,\phi_{-1,n}\rangle</math> so that <math display="inline">\psi(t)=\sqrt2 \sum_{n\in\Z} h_n\phi(2t-n).</math>
The second identity of the first pair is a [[Refinable function|refinement equation]] for the father wavelet φ. Both pairs of identities form the basis for the algorithm of the [[fast wavelet transform]].

From the multiresolution analysis derives the orthogonal decomposition of the space ''L''<sup>2</sup> as
<math display="block">L^2 = V_{j_0} \oplus W_{j_0} \oplus W_{j_0-1} \oplus W_{j_0-2} \oplus W_{j_0-3} \oplus \cdots</math>
For any signal or function <math>S\in L^2</math> this gives a representation in basis functions of the corresponding subspaces as
<math display="block">S = \sum_{k} c_{j_0,k}\phi_{j_0,k} + \sum_{j\le j_0}\sum_{k} d_{j,k}\psi_{j,k}</math>
where the coefficients are
<math display="block">c_{j_0,k} = \langle S,\phi_{j_0,k}\rangle </math> and
<math display="block">d_{j,k} = \langle S,\psi_{j,k}\rangle. </math>