Clifford wavelets

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

Specifically, denote these two families by and , where varies in the set of all dyadic cubes in and (the latter indicates that there correspond wavelets, left- and right-handed, respectively, to each fixed dyadic cube ; cf. also Wavelet analysis). In the simplest case (that of piecewise-constant, or Haar–Clifford, wavelets) they satisfy the following conditions:

1) , ;

2) , ;

3) and ;

4) .

Here is a fixed (typically accretive) Clifford-algebra-valued function in and the pairing is defined as

Due to the fact that the Clifford algebra-valued measure in no longer enjoys the usual translation and dilation properties of the Lebesgue measure, one cannot obtain families of functions as such via the familiar translation and dilation operations performed on some initial, fixed, function 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 -frames, i.e. that

for any square-integrable Clifford-algebra-valued function in .

References

[a1]	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
[a2]	M. Mitrea, "Clifford wavelets, singular integrals, and Hardy spaces" , Lecture Notes in Mathematics , 1575 , Springer (1994)