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The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110070/g1100702.png" />-dimensional Gauss (or Weierstrass) kernel
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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/g/g110/g110070/g1100706.png" /></td> </tr></table>
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with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110070/g1100707.png" /> a positive constant, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110070/g1100708.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110070/g1100709.png" />, is the [[Fundamental solution|fundamental solution]] of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110070/g11007010.png" />-dimensional [[Heat equation|heat equation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110070/g11007011.png" />. Moreover, this kernel is an approximate identity in that the Gauss–Weierstrass singular integral at the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110070/g11007012.png" />,
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The  $  n $-
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dimensional Gauss (or Weierstrass) kernel
  
<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/g/g110/g110070/g11007013.png" /></td> </tr></table>
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$$
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G ( x,y,t;D ) = ( 4 \pi Dt ) ^ {- n/2 } { \mathop{\rm exp} } \left ( - {
 +
\frac{1}{4Dt }
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} \left | {x - y } \right |  ^ {2} \right ) ,
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$$
  
satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110070/g11007014.png" /> almost everywhere, for example, whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110070/g11007015.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110070/g11007016.png" />; see [[#References|[a4]]]. Thus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110070/g11007017.png" /> is a solution of the heat equation for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110070/g11007018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110070/g11007019.png" /> having the initial  "temperature"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110070/g11007020.png" />.
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with  $  D $
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a positive constant,  $  t > 0 $,
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$  x,y \in \mathbf R  ^ {n} $,
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is the [[Fundamental solution|fundamental solution]] of the  $  n $-
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dimensional [[Heat equation|heat equation]]  $  u _ {t} = D \Delta u $.
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Moreover, this kernel is an approximate identity in that the Gauss–Weierstrass singular integral at the function  $  f $,
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 +
$$
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u ( x,t;D ) = \int\limits _ {\mathbf R  ^ {n} } {G ( x,y,t;D ) f ( y ) }  {dy } ,
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$$
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satisfies  $  {\lim\limits } _ {t \rightarrow 0 ^ {+}  } u ( x,t;D ) = f ( x ) $
 +
almost everywhere, for example, whenever $  \int _ {\mathbf R  ^ {n}  } {e ^ {- A | y |  ^ {2} } | {f ( y ) } | }  {dy } < \infty $
 +
for some $  A > 0 $;  
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see [[#References|[a4]]]. Thus $  u ( x,t;D ) $
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is a solution of the heat equation for $  0 < t < {1 / {( 4AD ) } } $,  
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$  x \in \mathbf R  ^ {n} $
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having the initial  "temperature"   $ f $.
  
 
In the theory of Markov processes (cf. [[Markov process|Markov process]]) the Gauss kernel gives the transition probability density of the Wiener–Lévy process (or of [[Brownian motion|Brownian motion]]). The semi-group property of the Gauss kernel
 
In the theory of Markov processes (cf. [[Markov process|Markov process]]) the Gauss kernel gives the transition probability density of the Wiener–Lévy process (or of [[Brownian motion|Brownian motion]]). The semi-group property of the Gauss kernel
  
<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/g/g110/g110070/g11007021.png" /></td> </tr></table>
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$$
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G ( x,z,t _ {1} + t _ {2} ;D ) = \int\limits _ {\mathbf R  ^ {n} } {G ( x,y,t _ {1} ;D ) G ( y,z,t _ {2} ;D ) }  {dy } ,
 +
$$
  
<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/g/g110/g110070/g11007022.png" /></td> </tr></table>
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$$
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t _ {1} ,t _ {2} > 0, \quad x,z \in \mathbf R  ^ {n} ,
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$$
  
 
is essential here.
 
is essential here.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P. Butzer,   R. Nessel,   "Fourier analysis and approximation" , '''I''' , Birkhäuser (1971)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Courant,   D. Hilbert,   "Methods of mathematical physics" , '''II''' , Wiley (1962)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> W. Feller,   "An introduction to probability theory and its applications" , '''2''' , Springer (1976) (Edition: Second)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> E.C. Titchmarsh,   "Introduction to the theory of Fourier integrals" , Clarendon Press (1937)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> K. Weierstrass,   "Ueber die analytische Darstellbarkeit sogenannter willkurlichen Functionen reeler Argumente" ''Berliner Sitzungsberichte'' (1985) pp. 633–639; 789–805</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P. Butzer, R. Nessel, "Fourier analysis and approximation", '''I''', Birkhäuser (1971)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Courant, D. Hilbert, "Methods of mathematical physics", '''II''', Wiley (1962)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> W. Feller, [[Feller, "An introduction to probability theory and its applications"|"An introduction to probability theory and its  applications"]], '''2''', Springer (1976) (Edition: Second)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> E.C. Titchmarsh, "Introduction to the theory of Fourier integrals", Clarendon Press (1937)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> K. Weierstrass, "Ueber die analytische Darstellbarkeit sogenannter willkurlichen Functionen reeler Argumente" ''Berliner Sitzungsberichte'' (1985) pp. 633–639; 789–805</TD></TR></table>

Latest revision as of 19:41, 5 June 2020


The $ n $- dimensional Gauss (or Weierstrass) kernel

$$ G ( x,y,t;D ) = ( 4 \pi Dt ) ^ {- n/2 } { \mathop{\rm exp} } \left ( - { \frac{1}{4Dt } } \left | {x - y } \right | ^ {2} \right ) , $$

with $ D $ a positive constant, $ t > 0 $, $ x,y \in \mathbf R ^ {n} $, is the fundamental solution of the $ n $- dimensional heat equation $ u _ {t} = D \Delta u $. Moreover, this kernel is an approximate identity in that the Gauss–Weierstrass singular integral at the function $ f $,

$$ u ( x,t;D ) = \int\limits _ {\mathbf R ^ {n} } {G ( x,y,t;D ) f ( y ) } {dy } , $$

satisfies $ {\lim\limits } _ {t \rightarrow 0 ^ {+} } u ( x,t;D ) = f ( x ) $ almost everywhere, for example, whenever $ \int _ {\mathbf R ^ {n} } {e ^ {- A | y | ^ {2} } | {f ( y ) } | } {dy } < \infty $ for some $ A > 0 $; see [a4]. Thus $ u ( x,t;D ) $ is a solution of the heat equation for $ 0 < t < {1 / {( 4AD ) } } $, $ x \in \mathbf R ^ {n} $ having the initial "temperature" $ f $.

In the theory of Markov processes (cf. Markov process) the Gauss kernel gives the transition probability density of the Wiener–Lévy process (or of Brownian motion). The semi-group property of the Gauss kernel

$$ G ( x,z,t _ {1} + t _ {2} ;D ) = \int\limits _ {\mathbf R ^ {n} } {G ( x,y,t _ {1} ;D ) G ( y,z,t _ {2} ;D ) } {dy } , $$

$$ t _ {1} ,t _ {2} > 0, \quad x,z \in \mathbf R ^ {n} , $$

is essential here.

References

[a1] P. Butzer, R. Nessel, "Fourier analysis and approximation", I, Birkhäuser (1971)
[a2] R. Courant, D. Hilbert, "Methods of mathematical physics", II, Wiley (1962)
[a3] W. Feller, "An introduction to probability theory and its applications", 2, Springer (1976) (Edition: Second)
[a4] E.C. Titchmarsh, "Introduction to the theory of Fourier integrals", Clarendon Press (1937)
[a5] K. Weierstrass, "Ueber die analytische Darstellbarkeit sogenannter willkurlichen Functionen reeler Argumente" Berliner Sitzungsberichte (1985) pp. 633–639; 789–805
How to Cite This Entry:
Gauss kernel. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gauss_kernel&oldid=13441
This article was adapted from an original article by R. Kerman (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article