View source for Clifford wavelets

A pair of families of Clifford algebra-valued functions satisfying appropriate smoothness, size, cancellation, and orthogonality conditions (cf. also [[Clifford algebra|Clifford algebra]]).

Specifically, denote these two families by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110270/c1102701.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110270/c1102702.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110270/c1102703.png" /> varies in the set of all dyadic cubes in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110270/c1102704.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110270/c1102705.png" /> (the latter indicates that there correspond <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110270/c1102706.png" /> wavelets, left- and right-handed, respectively, to each fixed dyadic cube <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110270/c1102707.png" />; cf. also [[Wavelet analysis|Wavelet analysis]]). In the simplest case (that of piecewise-constant, or Haar–Clifford, wavelets) they satisfy the following conditions:

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110270/c1102708.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110270/c1102709.png" />;

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110270/c11027010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110270/c11027011.png" />;

3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110270/c11027012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110270/c11027013.png" />;

4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110270/c11027014.png" />.

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110270/c11027015.png" /> is a fixed (typically accretive) Clifford-algebra-valued function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110270/c11027016.png" /> and the pairing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110270/c11027017.png" /> is defined as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110270/c11027018.png" /></td> </tr></table>

Due to the fact that the Clifford algebra-valued measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110270/c11027019.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110270/c11027020.png" /> no longer enjoys the usual translation and dilation properties of the [[Lebesgue measure|Lebesgue measure]], one cannot obtain families of functions as such via the familiar translation and dilation operations performed on some initial, fixed, function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110270/c11027021.png" /> as in the case of ordinary wavelets. However, as the above conditions suggest, everything happens as if one could.

For many applications it is crucial that such families are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110270/c11027023.png" />-frames, i.e. that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110270/c11027024.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110270/c11027025.png" /></td> </tr></table>

for any square-integrable Clifford-algebra-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110270/c11027026.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110270/c11027027.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Auscher,   Ph. Tchamitchian,   "Bases d'ondelettes sur des courbes corde-arc, noyau de Cauchy et espaces de Hardy associés"  ''Rev. Mat. Iberoamericana'' , '''5'''  (1989)  pp. 139–170</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Mitrea,   "Clifford wavelets, singular integrals, and Hardy spaces" , ''Lecture Notes in Mathematics'' , '''1575''' , Springer  (1994)</TD></TR></table>