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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the singularities of continuous Legendre transforms
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by Gilbert G. Walter and Ahmed I. Zayed PDF
Proc. Amer. Math. Soc. 97 (1986), 673-681 Request permission

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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Additional Information
  • © Copyright 1986 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 97 (1986), 673-681
  • MSC: Primary 44A15
  • DOI: https://doi.org/10.1090/S0002-9939-1986-0845986-9
  • MathSciNet review: 845986