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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The analytic properties of continuous Legendre transform of a function holomorphic in an elliptical neighborhood of the real interval are investigated. It is shown to be an entire function of exponential type whose Borel transform has a singularity at if and only if has one at where . 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 in terms of .

**[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)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
44A15

Retrieve articles in all journals with MSC: 44A15

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