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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the singularities of continuous Legendre transforms


Authors: Gilbert G. Walter and Ahmed I. Zayed
Journal: Proc. Amer. Math. Soc. 97 (1986), 673-681
MSC: Primary 44A15
MathSciNet review: 845986
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Abstract: The analytic properties of continuous Legendre transform $ F(\lambda )$ of a function $ f(t)$ holomorphic in an elliptical neighborhood of the real interval $ [ - 1,1]$ are investigated. It is shown to be an entire function of exponential type whose Borel transform $ g(z)$ has a singularity at $ {z_0}$ if and only if $ f(t)$ has one at $ {t_0}$ where $ {z_0} = \cosh {t_0}$. The proof involves a modification of "Hadamard's argument" on multiplication of singularities. The result may also be interpreted as a statement about the second continuous Legendre transform which gives $ f(t)$ in terms of $ F(\lambda )$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1986-0845986-9
PII: S 0002-9939(1986)0845986-9
Keywords: Continuous Legendre transforms, singularities, analytic continuation
Article copyright: © Copyright 1986 American Mathematical Society