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The Appell transform and the semigroup property for temperatures

Part of: Nontrigonometric harmonic analysis Parabolic equations and systems

Published online by Cambridge University Press:  09 April 2009

Elizabeth Kochneff  and
Yoram Sagher
Elizabeth Kochneff
Affiliation:
University of Illinois at ChicagoChicago, Illinois 60680, USA
Yoram Sagher
Affiliation:
University of Illinois at ChicagoChicago, Illinois 60680, USA
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Abstract

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We prove that if u(x, t) is a solution of the one dimensional heat equation and if A u(x, t) is its Appell transform, then u(x, t) has the semi-group (Huygens) property in a domain D if and only if A u(x, t) has the semi-group property in a dual region. We apply this result to simplify and extend some results of Rosenbloom and Widder.

MSC classification

Secondary: 35K05: Heat equation 42C99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 60 , Issue 1 , February 1996 , pp. 109 - 117
Copyright
Copyright © Australian Mathematical Society 1996

References

[1]Appell, P., ‘Sur l'équation ∂2z/∂x2 − ∂z/∂y et la théorie de la chaleur’, J. Math. Pures Appl. 8 (1892), 186–216.Google Scholar
[2]Rosenbloom, P. C. and Widder, D. V., ‘Expansions in terms of heat polynomials and associated functions’, Trans. Amer. Math. Soc. 92 (1959), 220–266.Google Scholar
[3]Widder, D. V., The heat equation (Academic Press, San Diego, 1975).Google Scholar