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 $ \{ \Theta ^ {\textrm{ l } } _ {Q,j } \} _ {Q,j } $ and $ \{ \Theta ^ {\textrm{ r } } _ {Q,j } \} _ {Q,j } $, where $ Q $ varies in the set of all dyadic cubes in $ \mathbf R ^ {m} $ and $ j = 1 \dots 2 ^ {m} - 1 $( the latter indicates that there correspond $ 2 ^ {m} - 1 $ wavelets, left- and right-handed, respectively, to each fixed dyadic cube $ Q $; cf. also Wavelet analysis). In the simplest case (that of piecewise-constant, or Haar–Clifford, wavelets) they satisfy the following conditions:

1) $ \supp \Theta ^ {\textrm{ l } } _ {Q,j } $, $ \supp \Theta ^ {\textrm{ r } } _ {Q,j } \subseteq Q $;

2) $ | {\Theta ^ {\textrm{ l } } _ {Q,j } } | $, $ | {\Theta ^ {\textrm{ r } } _ {Q,j } } | \leq C | Q | ^ {- {1 / 2 } } $;

3) $ \langle {\Theta ^ {\textrm{ l } } _ {Q,j } ,1 } \rangle _ {b} = 0 $ and $ \langle {1, \Theta ^ {\textrm{ r } } _ {Q,j } } \rangle _ {b} = 0 $;

4) $ \langle {\Theta ^ {\textrm{ l } } _ {Q,j } , \Theta ^ {\textrm{ r } } _ {Q ^ \prime ,j ^ \prime } } \rangle _ {b} = \delta _ {QQ ^ \prime } \delta _ {ij } e _ {0} $.

Here $ b $ is a fixed (typically accretive) Clifford-algebra-valued function in $ \mathbf R ^ {m} $ and the pairing $ \langle {\cdot, \cdot } \rangle _ {b} $ is defined as

$$ \left \langle {f _ {1} ,f _ {2} } \right \rangle _ {b} = \int\limits _ {\mathbf R ^ {m} } {f _ {1} ( x ) b ( x ) f _ {2} ( x ) } {dx } . $$

Due to the fact that the Clifford algebra-valued measure $ b ( x ) dx $ in $ \mathbf R ^ {m} $ 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 $ \Theta $ 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 $ L _ {2} $- frames, i.e. that

$$ f = \sum \Theta ^ {\textrm{ r } } _ {Q,j } \left \langle {\Theta ^ {\textrm{ l } } _ {Q,j } ,f } \right \rangle _ {b} = \sum \left \langle {f, \Theta ^ {\textrm{ r } } _ {Q,j } } \right \rangle _ {b} \Theta ^ {\textrm{ l } } _ {Q,j } , $$

$$ \left \| f \right \| ^ {2} _ {L ^ {2} ( \mathbf R ^ {m} ) } \approx \sum \left | {\left \langle {\Theta ^ {\textrm{ l } } _ {Q,j } ,f } \right \rangle _ {b} } \right | ^ {2} \approx \sum \left | {\left \langle {f, \Theta ^ {\textrm{ r } } _ {Q,j } } \right \rangle _ {b} } \right | ^ {2} $$

for any square-integrable Clifford-algebra-valued function $ f $ in $ \mathbf R ^ {m} $.

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)