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Analytic continuation and identities involving heat, Poisson, wave and Bessel kernels
Author(s) -
Jorgenson Jay,
Lang Serge
Publication year - 2003
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200310086
Subject(s) - heat kernel , mathematics , bessel function , poisson kernel , continuation , poisson distribution , riemannian manifold , kernel (algebra) , mathematical analysis , gravitational singularity , meromorphic function , manifold (fluid mechanics) , analytic continuation , bessel process , pure mathematics , orthogonal polynomials , gegenbauer polynomials , statistics , mechanical engineering , classical orthogonal polynomials , computer science , engineering , programming language
In this article we use classical formulas involving the K –Bessel function in two variables to express the Poisson kernel on a Riemannian manifold in terms of the heat kernel. We then use the small time asymptotics of the heat kernel on certain Riemannian manifolds to obtain a meromorphic continuation of the associated Poisson kernel to all values of complex time with identifiable singularities. This result reproves in a different setting by different means a well–known theorem due to Duistermaat and Guillemin [DG 75]. Also, we develop analytic expressions for the heat kernel beyond asymptotic expansions. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)