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''of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097060/w0970601.png" /> on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097060/w0970602.png" />''
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The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097060/w0970603.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097060/w0970604.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097060/w0970605.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097060/w0970606.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097060/w0970607.png" /> are the Rademacher functions (cf. [[Rademacher system|Rademacher system]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097060/w0970608.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097060/w0970609.png" />, is the binary representation of the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097060/w09706010.png" />. This system was defined and studied by J.L. Walsh [[#References|[1]]], but already in 1900 J.A. Barrett studied functions of this system in questions connected with the distribution of electrons on open conducting curves. In connection with this theory another definition of Walsh functions is preferred. Namely, if
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{{TEX|done}}
  
<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/w/w097/w097060/w09706011.png" /></td> </tr></table>
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''of functions  $  \{ W _ {n} \} $
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on the interval  $  [ 0, 1] $''
  
then the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097060/w09706012.png" /> are defined by the following recurrence formulas:
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The functions $  W _ {0} ( x) \equiv 1 $
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and  $  W _ {n} ( x) = r _ {\nu _ {1}  } ( x) \dots r _ {\nu _ {m}  } ( x) $
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for  $  n \geq  1 $,
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where  $  r _ {k} ( x) = \mathop{\rm sign}  \sin  2 ^ {k + 1 } \pi x $,
 +
$  k = 0, 1 \dots $
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are the Rademacher functions (cf. [[Rademacher system|Rademacher system]]) and  $  n = 2 ^ {\nu _ {1} } + {} \dots + 2 ^ {\nu _ {m} } $,
 +
$  \nu _ {1} > \dots > \nu _ {m} $,
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is the binary representation of the number  $  n \geq  1 $.  
 +
This system was defined and studied by J.L. Walsh [[#References|[1]]], but already in 1900 J.A. Barrett studied functions of this system in questions connected with the distribution of electrons on open conducting curves. In connection with this theory another definition of Walsh functions is preferred. Namely, if
  
<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/w/w097/w097060/w09706013.png" /></td> </tr></table>
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$$
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W _ {0}  ^ {*} ( x)  = \
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\left \{
  
<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/w/w097/w097060/w09706014.png" /></td> </tr></table>
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then the functions  $  W _ {n}  ^ {*} ( x) $
 +
are defined by the following recurrence formulas:
  
The systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097060/w09706015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097060/w09706016.png" /> differ only in their ordering in the ranges <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097060/w09706017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097060/w09706018.png" />. For example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097060/w09706019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097060/w09706020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097060/w09706021.png" />, etc. The index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097060/w09706022.png" /> of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097060/w09706023.png" /> corresponds to the number of changes of sign of this function in the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097060/w09706024.png" />, i.e. it is the analogue to doubling the frequency of a sinusoidal function. The Walsh system is a complete orthonormal system on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097060/w09706025.png" /> and it may be considered as a natural completion of the [[Rademacher system|Rademacher system]].
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$$
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W _ {2j + p }  ^ {*} ( x)  = \
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W _ {j}  ^ {*} ( 2x) + (- 1) ^ {j + p } W _ {j}  ^ {*} ( 2x - 1),
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$$
  
The Walsh system forms a commutative multiplicative group, with the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097060/w09706026.png" /> as unit element, while each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097060/w09706027.png" /> is its own inverse.
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$$
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p  =  0, 1; \  j  =  0, 1 , .  .  . .
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$$
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 +
The systems  $  \{ W _ {n} \} $
 +
and  $  \{ W _ {n}  ^ {*} \} $
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differ only in their ordering in the ranges  $  2  ^ {m} \leq  n \leq  2 ^ {m + 1 } - 1 $,
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$  m = 1, 2 , .  .  . $.
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For example,  $  W _ {2  ^ {m}  }  ^ {*} = W _ {3 \cdot 2 ^ {m - 1 }  } $,
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$  W _ {2 ^ {m - 1 }  - 1 }  ^ {*} = W _ {2  ^ {m}  } $,
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$  W _ {2 ^ {m + 1 }  - 2 }  ^ {*} = W _ {2  ^ {m}  + 1 } $,
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etc. The index  $  k $
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of the function  $  W _ {k}  ^ {*} $
 +
corresponds to the number of changes of sign of this function in the interval  $  ( 0, 1) $,
 +
i.e. it is the analogue to doubling the frequency of a sinusoidal function. The Walsh system is a complete orthonormal system on the interval  $  [ 0, 1] $
 +
and it may be considered as a natural completion of the [[Rademacher system|Rademacher system]].
 +
 
 +
The Walsh system forms a commutative multiplicative group, with the function $  W _ {0} $
 +
as unit element, while each $  W _ {k} $
 +
is its own inverse.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.L. Walsh,  "A closed set of normal orthogonal functions"  ''Amer. J. Math.'' , '''45'''  (1923)  pp. 5–24</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F.F. Fowle,  ''Trans. AJEE'' , '''23'''  (1905)  pp. 659–687</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.J. Fine,  "On the Walsh functions"  ''Trans. Amer. Math. Soc.'' , '''65'''  (1949)  pp. 372–414</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Kaczmarz,  H. Steinhaus,  "Theorie der Orthogonalreihen" , Chelsea, reprint  (1951)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  H.F. Harmut,  "Transmission of information by orthogonal functions" , Springer  (1972)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.L. Walsh,  "A closed set of normal orthogonal functions"  ''Amer. J. Math.'' , '''45'''  (1923)  pp. 5–24</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F.F. Fowle,  ''Trans. AJEE'' , '''23'''  (1905)  pp. 659–687</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.J. Fine,  "On the Walsh functions"  ''Trans. Amer. Math. Soc.'' , '''65'''  (1949)  pp. 372–414</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Kaczmarz,  H. Steinhaus,  "Theorie der Orthogonalreihen" , Chelsea, reprint  (1951)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  H.F. Harmut,  "Transmission of information by orthogonal functions" , Springer  (1972)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Golubov,  A. Efimov,  V. Skvortsov,  "Walsh series and transforms" , Kluwer  (1987)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Golubov,  A. Efimov,  V. Skvortsov,  "Walsh series and transforms" , Kluwer  (1987)  (Translated from Russian)</TD></TR></table>

Revision as of 08:28, 6 June 2020


of functions $ \{ W _ {n} \} $ on the interval $ [ 0, 1] $

The functions $ W _ {0} ( x) \equiv 1 $ and $ W _ {n} ( x) = r _ {\nu _ {1} } ( x) \dots r _ {\nu _ {m} } ( x) $ for $ n \geq 1 $, where $ r _ {k} ( x) = \mathop{\rm sign} \sin 2 ^ {k + 1 } \pi x $, $ k = 0, 1 \dots $ are the Rademacher functions (cf. Rademacher system) and $ n = 2 ^ {\nu _ {1} } + {} \dots + 2 ^ {\nu _ {m} } $, $ \nu _ {1} > \dots > \nu _ {m} $, is the binary representation of the number $ n \geq 1 $. This system was defined and studied by J.L. Walsh [1], but already in 1900 J.A. Barrett studied functions of this system in questions connected with the distribution of electrons on open conducting curves. In connection with this theory another definition of Walsh functions is preferred. Namely, if

$$ W _ {0} ^ {*} ( x) = \ \left \{ then the functions $ W _ {n} ^ {*} ( x) $ are defined by the following recurrence formulas: $$ W _ {2j + p } ^ {*} ( x) = \ W _ {j} ^ {*} ( 2x) + (- 1) ^ {j + p } W _ {j} ^ {*} ( 2x - 1), $$ $$ p = 0, 1; \ j = 0, 1 , . . . . $$

The systems $ \{ W _ {n} \} $ and $ \{ W _ {n} ^ {*} \} $ differ only in their ordering in the ranges $ 2 ^ {m} \leq n \leq 2 ^ {m + 1 } - 1 $, $ m = 1, 2 , . . . $. For example, $ W _ {2 ^ {m} } ^ {*} = W _ {3 \cdot 2 ^ {m - 1 } } $, $ W _ {2 ^ {m - 1 } - 1 } ^ {*} = W _ {2 ^ {m} } $, $ W _ {2 ^ {m + 1 } - 2 } ^ {*} = W _ {2 ^ {m} + 1 } $, etc. The index $ k $ of the function $ W _ {k} ^ {*} $ corresponds to the number of changes of sign of this function in the interval $ ( 0, 1) $, i.e. it is the analogue to doubling the frequency of a sinusoidal function. The Walsh system is a complete orthonormal system on the interval $ [ 0, 1] $ and it may be considered as a natural completion of the Rademacher system.

The Walsh system forms a commutative multiplicative group, with the function $ W _ {0} $ as unit element, while each $ W _ {k} $ is its own inverse.

References

[1] J.L. Walsh, "A closed set of normal orthogonal functions" Amer. J. Math. , 45 (1923) pp. 5–24
[2] F.F. Fowle, Trans. AJEE , 23 (1905) pp. 659–687
[3] N.J. Fine, "On the Walsh functions" Trans. Amer. Math. Soc. , 65 (1949) pp. 372–414
[4] S. Kaczmarz, H. Steinhaus, "Theorie der Orthogonalreihen" , Chelsea, reprint (1951)
[5] H.F. Harmut, "Transmission of information by orthogonal functions" , Springer (1972)

Comments

References

[a1] B. Golubov, A. Efimov, V. Skvortsov, "Walsh series and transforms" , Kluwer (1987) (Translated from Russian)
How to Cite This Entry:
Walsh system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Walsh_system&oldid=49171
This article was adapted from an original article by A.V. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article