Gauss kernel
From Encyclopedia of Mathematics
The -dimensional Gauss (or Weierstrass) kernel
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with a positive constant,
,
, is the fundamental solution of the
-dimensional heat equation
. Moreover, this kernel is an approximate identity in that the Gauss–Weierstrass singular integral at the function
,
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satisfies almost everywhere, for example, whenever
for some
; see [a4]. Thus
is a solution of the heat equation for
,
having the initial "temperature"
.
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
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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=25928
Gauss kernel. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gauss_kernel&oldid=25928
This article was adapted from an original article by R. Kerman (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article