Editing Haar wavelet (section)

==Haar wavelet properties==
The Haar wavelet has several notable properties:

{{ordered list

| Any continuous real function with compact support can be approximated uniformly by [[linear combination]]s of <math>\varphi(t),\varphi(2t),\varphi(4t),\dots,\varphi(2^n t),\dots</math> and their shifted functions. This extends to those function spaces where any function therein can be approximated by continuous functions.

| Any continuous real function on [0,&nbsp;1] can be approximated uniformly on [0,&nbsp;1] by linear combinations of the constant function&nbsp;'''1''', <math>\psi(t),\psi(2t),\psi(4t),\dots,\psi(2^n t),\dots</math> and their shifted functions.<ref>As opposed to the preceding statement, this fact is not obvious: see p.&nbsp;363 in {{harvtxt|Haar|1910}}.</ref>

| [[Orthogonality]] in the form
: <math>
 \int_{-\infty}^{\infty}2^{(n+n_1)/2}\psi(2^n t-k)\psi(2^{n_1} t - k_1)\, dt = \delta_{nn_1}\delta_{kk_1}.
 </math>
Here, <math>\delta_{ij}</math> represents the [[Kronecker delta]]. The [[dual function]] of ψ(''t'') is ψ(''t'') itself.

| Wavelet/scaling functions with different scale ''n'' have a functional relationship:<ref>{{cite book |last=Vidakovic |first=Brani |title=Statistical Modeling by Wavelets |series=Wiley Series in Probability and Statistics |year=2010 |edition=2 |doi=10.1002/9780470317020 |pages=60, 63|isbn=9780470317020 }}</ref> since
:<math>
\begin{align}
  \varphi(t) &= \varphi(2t)+\varphi(2t-1)\\[.2em]
  \psi(t) &= \varphi(2t)-\varphi(2t-1),
\end{align}</math>
it follows that coefficients of scale ''n'' can be calculated by coefficients of scale ''n+1'':<br/>
If <math> \chi_w(k, n)= 2^{n/2}\int_{-\infty}^\infty x(t)\varphi(2^nt-k)\, dt</math><br/>
and <math> \Chi_w(k, n)= 2^{n/2}\int_{-\infty}^\infty x(t)\psi(2^nt-k)\, dt</math><br/>
then
: <math> \chi_w(k,n)= 2^{-1/2} \bigl( \chi_w(2k,n+1)+\chi_w(2k+1,n+1) \bigr)</math>
: <math> \Chi_w(k,n)= 2^{-1/2} \bigl( \chi_w(2k,n+1)-\chi_w(2k+1,n+1) \bigr).</math>

}}
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