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A pair of families of Clifford algebra-valued functions satisfying appropriate smoothness, size, cancellation, and orthogonality conditions (cf. also [[Clifford algebra|Clifford algebra]]).
 
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:
+
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|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" />;
+
1) $  \supp  \Theta ^ {\textrm{ l } } _ {Q,j }  $,
 +
$  \supp  \Theta ^ {\textrm{ r } } _ {Q,j }  \subseteq Q $;
  
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" />;
+
2) $  | {\Theta ^ {\textrm{ l } } _ {Q,j }  } | $,
 +
$  | {\Theta ^ {\textrm{ r } } _ {Q,j }  } | \leq  C | Q | ^ {- {1 / 2 } } $;
  
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" />;
+
3) $  \langle  {\Theta ^ {\textrm{ l } } _ {Q,j }  ,1 } \rangle _ {b} = 0 $
 +
and $  \langle  {1, \Theta ^ {\textrm{ r } } _ {Q,j }  } \rangle _ {b} = 0 $;
  
4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110270/c11027014.png" />.
+
4) $  \langle  {\Theta ^ {\textrm{ l } } _ {Q,j }  , \Theta ^ {\textrm{ r } } _ {Q  ^  \prime  ,j  ^  \prime  } } \rangle _ {b} = \delta _ {QQ  ^  \prime  } \delta _ {ij }  e _ {0} $.
  
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
+
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
  
<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>
+
$$
 +
\left \langle  {f _ {1} ,f _ {2} } \right \rangle _ {b} = \int\limits _ {\mathbf R  ^ {m} } {f _ {1} ( x )  b ( x )  f _ {2} ( x ) }  {dx } .
 +
$$
  
Due to the fact that the Clifford algebra-valued measure <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.
+
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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110270/c11027023.png" />-frames, i.e. that
+
For many applications it is crucial that such families are $  L _ {2} $-
 +
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>
+
$$
 +
f = \sum \Theta ^ {\textrm{ r } } _ {Q,j }  \left \langle  {\Theta ^ {\textrm{ l } } _ {Q,j }  ,f } \right \rangle _ {b} = \sum \left \langle  {f, \Theta ^ {\textrm{ r } } _ {Q,j }  } \right \rangle _ {b} \Theta ^ {\textrm{ l } } _ {Q,j }  ,
 +
$$
  
<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>
+
$$
 +
\left \| f \right \|  ^ {2} _ {L  ^ {2}  ( \mathbf R  ^ {m} ) } \approx \sum \left | {\left \langle  {\Theta ^ {\textrm{ l } } _ {Q,j }  ,f } \right \rangle _ {b} } \right |  ^ {2} \approx \sum \left | {\left \langle  {f, \Theta ^ {\textrm{ r } } _ {Q,j }  } \right \rangle _ {b} } \right |  ^ {2}
 +
$$
  
for any square-integrable Clifford-algebra-valued function <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" />.
+
for any square-integrable Clifford-algebra-valued function $  f $
 +
in $  \mathbf R  ^ {m} $.
  
 
====References====
 
====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>
 
<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>

Latest revision as of 17:44, 4 June 2020


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)
How to Cite This Entry:
Clifford wavelets. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Clifford_wavelets&oldid=46361
This article was adapted from an original article by M. Mitrea (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article