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A [[Harmonic function|harmonic function]] such that the [[Laplace operator|Laplace operator]] acting on separate groups of independent variables vanishes. More precisely: A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m0652701.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m0652702.png" />, of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m0652703.png" /> in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m0652704.png" /> of the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m0652705.png" /> is called a multiharmonic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m0652706.png" /> if there exist natural numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m0652707.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m0652708.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m0652709.png" />, such that the following identities hold throughout <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m06527010.png" />:
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<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/m/m065/m065270/m06527011.png" /></td> </tr></table>
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<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/m/m065/m065270/m06527012.png" /></td> </tr></table>
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A [[Harmonic function|harmonic function]] such that the [[Laplace operator|Laplace operator]] acting on separate groups of independent variables vanishes. More precisely: A function  $  u = u ( x _ {1} \dots x _ {n} ) $,
 +
$  n \geq  2 $,
 +
of class  $  C  ^ {2} $
 +
in a domain  $  D $
 +
of the Euclidean space  $  \mathbf R  ^ {n} $
 +
is called a multiharmonic function in  $  D $
 +
if there exist natural numbers  $  n _ {1} \dots n _ {k} $,
 +
$  n _ {1} + \dots + n _ {k} = n $,
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$  n \geq  k \geq  2 $,
 +
such that the following identities hold throughout  $  D $:
  
An important proper subclass of the class of multiharmonic functions consists of the pluriharmonic functions (cf. [[Pluriharmonic function|Pluriharmonic function]]), for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m06527013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m06527014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m06527015.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m06527016.png" />, and which also satisfy certain additional conditions.
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$$
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\sum _ { \nu = 1 } ^ { {n _ 1 } }
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 +
\frac{\partial  ^ {2} u }{\partial  x _  \nu  ^ {2} }
 +
  =  0,
 +
$$
 +
 
 +
$$
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\sum _ { \nu = n _ {1} + 1 } ^ { {n _ 1 } + n _ {2} }
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\frac{\partial  ^ {2} u }{\partial  x _  \nu  ^ {2} }
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  =  0 \dots \sum
 +
_ {\nu = n _ {1} + \dots + n _ {k - 1 }  + 1 } ^ { n } 
 +
\frac{\partial  ^ {2} u }{\partial  x _  \nu  ^ {2} }
 +
  =  0.
 +
$$
 +
 
 +
An important proper subclass of the class of multiharmonic functions consists of the pluriharmonic functions (cf. [[Pluriharmonic function|Pluriharmonic function]]), for which $  n = 2m $,  
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$  n _ {j} = 2 $,  
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$  j = 1 \dots m $,  
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i.e. $  k = m $,  
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and which also satisfy certain additional conditions.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.M. Stein,  G. Weiss,  "Introduction to harmonic analysis on Euclidean spaces" , Princeton Univ. Press  (1971)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.M. Stein,  G. Weiss,  "Introduction to harmonic analysis on Euclidean spaces" , Princeton Univ. Press  (1971)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Multiharmonic functions are also called multiply harmonic functions. It was shown by P. Lelong that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m06527017.png" /> is separately harmonic, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m06527018.png" /> is harmonic as a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m06527019.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m06527020.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m06527021.png" />) while the other variables remain fixed, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065270/m06527022.png" /> is multiharmonic. A different proof is due to J. Siciak. See [[#References|[a1]]].
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Multiharmonic functions are also called multiply harmonic functions. It was shown by P. Lelong that if $  u $
 +
is separately harmonic, that is, $  u $
 +
is harmonic as a function of $  x _ {n _ {j}  + 1 } \dots x _ {n _ {j+} 1 }  $(
 +
$  j= 0 \dots k $;  
 +
$  n _ {0} = 0 $)  
 +
while the other variables remain fixed, then $  u $
 +
is multiharmonic. A different proof is due to J. Siciak. See [[#References|[a1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Hervé,  "Analytic and plurisubharmonic functions" , ''Lect. notes in math.'' , '''198''' , Springer  (1971)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Hervé,  "Analytic and plurisubharmonic functions" , ''Lect. notes in math.'' , '''198''' , Springer  (1971)</TD></TR></table>

Latest revision as of 08:01, 6 June 2020


A harmonic function such that the Laplace operator acting on separate groups of independent variables vanishes. More precisely: A function $ u = u ( x _ {1} \dots x _ {n} ) $, $ n \geq 2 $, of class $ C ^ {2} $ in a domain $ D $ of the Euclidean space $ \mathbf R ^ {n} $ is called a multiharmonic function in $ D $ if there exist natural numbers $ n _ {1} \dots n _ {k} $, $ n _ {1} + \dots + n _ {k} = n $, $ n \geq k \geq 2 $, such that the following identities hold throughout $ D $:

$$ \sum _ { \nu = 1 } ^ { {n _ 1 } } \frac{\partial ^ {2} u }{\partial x _ \nu ^ {2} } = 0, $$

$$ \sum _ { \nu = n _ {1} + 1 } ^ { {n _ 1 } + n _ {2} } \frac{\partial ^ {2} u }{\partial x _ \nu ^ {2} } = 0 \dots \sum _ {\nu = n _ {1} + \dots + n _ {k - 1 } + 1 } ^ { n } \frac{\partial ^ {2} u }{\partial x _ \nu ^ {2} } = 0. $$

An important proper subclass of the class of multiharmonic functions consists of the pluriharmonic functions (cf. Pluriharmonic function), for which $ n = 2m $, $ n _ {j} = 2 $, $ j = 1 \dots m $, i.e. $ k = m $, and which also satisfy certain additional conditions.

References

[1] E.M. Stein, G. Weiss, "Introduction to harmonic analysis on Euclidean spaces" , Princeton Univ. Press (1971)

Comments

Multiharmonic functions are also called multiply harmonic functions. It was shown by P. Lelong that if $ u $ is separately harmonic, that is, $ u $ is harmonic as a function of $ x _ {n _ {j} + 1 } \dots x _ {n _ {j+} 1 } $( $ j= 0 \dots k $; $ n _ {0} = 0 $) while the other variables remain fixed, then $ u $ is multiharmonic. A different proof is due to J. Siciak. See [a1].

References

[a1] M. Hervé, "Analytic and plurisubharmonic functions" , Lect. notes in math. , 198 , Springer (1971)
How to Cite This Entry:
Multiharmonic function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multiharmonic_function&oldid=47925
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article