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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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Keywords: Continuous Legendre transforms, singularities, analytic continuation
Article copyright: © Copyright 1986 American Mathematical Society