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Gauss kernel

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The -dimensional Gauss (or Weierstrass) kernel

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 ,

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

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
This article was adapted from an original article by R. Kerman (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article