## 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 $F(\lambda )$.## References

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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