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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
DOI: https://doi.org/10.1090/S0002-9939-1986-0845986-9
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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  • [1] P. Butzer, R. Stens and M. Wehrens, The continuous Legendre transform: its inverse transform and applications, Internat. J. Math. Sci. 3 (1980), 47-67. MR 576629 (81h:44002)
  • [2] A. Erdelyi, Higher transcendental functions, vol. 1, McGraw-Hill, New York, 1953.
  • [3] R. Gilbert, Integral operator methods in bi-axially symmetric potential theory, Contrib. Differential Equations 2 (1963), 441-456. MR 0156998 (28:239)
  • [4] -, Bergman's integral operator method in generalized axially symmetric potential theory, J. Math. Phys. 5 (1964), 983-997. MR 0165131 (29:2420)
  • [5] -, Function theoretic methods in partial differential equations, Academic Press, New York, 1969. MR 0241789 (39:3127)
  • [6] R. Gilbert and H. Howard, A generalization of a theorem of Nehari, Bull. Amer. Math. Soc. 72 (1966), 37-43. MR 0183853 (32:1329)
  • [7] B. Ja. Levin, Distribution of zeros of entire functions, Trans. Math. Monographs, vol. 5, Amer. Math. Soc., Providence, R.I., 1964. MR 0156975 (28:217)
  • [8] Z. Nehari, On the singularities of Legendre expansions, J. Rational Mech. Anal. 5 (1956), 987-992. MR 0080747 (18:293d)
  • [9] G. Szegö, On the singularities of zonal harmonic expansions, J. Rational Mech. Anal. 3 (1954), 561-564. MR 0062880 (16:34b)
  • [10] G. Walter, On real singularities of Legendre expansions, Proc. Amer. Math. Soc. 19 (1968), 1407-1412. MR 0257635 (41:2285)
  • [11] A. Zayed, On the singularities of Gegenbauer (ultraspherical) expansions, Trans. Amer. Math. Soc. 262 (1980), 487-503. MR 586730 (82c:33016)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1986-0845986-9
Keywords: Continuous Legendre transforms, singularities, analytic continuation
Article copyright: © Copyright 1986 American Mathematical Society

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