Fourier transforms having only real zeros

Cardon, David

doi:10.1090/S0002-9939-04-07677-4

Fourier transforms having only real zeros

Author: David A. Cardon
Journal: Proc. Amer. Math. Soc. 133 (2005), 1349-1356
MSC (2000): Primary 42A38, 30C15
DOI: https://doi.org/10.1090/S0002-9939-04-07677-4
Published electronically: October 18, 2004
MathSciNet review: 2111941
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Abstract: Let be a real entire function of order less than with only real zeros. Then we classify certain distribution functions such that the Fourier transform $H(z)=\int_{-\infty}^{\infty}G(it)e^{izt}\,dF(t)$ has only real zeros.

1. Lars V. Ahlfors, Complex analysis: An introduction to the theory of analytic functions of one complex variable, third ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York, 1978. MR 0510197 (80c:30001)
2. David A. Cardon, Convolution operators and zeros of entire functions, Proc. Amer. Math. Soc. 130 (2002), no. 6, 1725-1734. MR 1887020 (2002m:30006)
3. David A. Cardon, Sums of exponential functions having only real zeros, Manuscripta Math. (to appear).
4. David A. Cardon and Pace P. Nielsen, Convolution operators and entire functions with simple zeros, Number theory for the millennium, I (Urbana, IL, 2000), A K Peters, Natick, MA, 2002, pp. 183-196. MR 1956225 (2003m:30012)
5. Louis de Branges, Hilbert Spaces of Entire Functions, Prentice-Hall Inc., Englewood Cliffs, N.J., 1968. MR 0229011 (37:4590)
6. M. L. Eaton, A probability inequality for linear combinations of bounded random variables, Ann. Stat. 2 (1974), 609-614.
7. Martin Eisen, Introduction to mathematical probability theory, Prentice-Hall Inc., Englewood Cliffs, N.J., 1969. MR 0258078 (41:2725)
8. T. D. Lee and C. N. Yang, Statistical theory of equations of state and phase transitions. II. Lattice gas and Ising model, Physical Rev. (2) 87 (1952), 410-419. MR 0053029 (14:711c)
9. Michel Loève, Probability theory, third ed., D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1963. MR 0203748 (34:3596)
10. Iosif Pinelis, Extremal probabilistic problems and Hotelling's ${T}\sp 2$ test under a symmetry condition, Ann. Statist. 22 (1994), no. 1, 357-368. MR 1272088 (95m:62115)
11. George Pólya, On the zeros of an integral function represented by Fourier's integral, Messenger of Math. 52 (1923), 185-188.
12. George Pólya, Bemerkung über die Integraldarstellung der Riemannschen $\xi$ -Funktion, Acta Math. 48 (1926), 305-317.
13. George Pólya, On the zeros of certain trigonometric integrals, J. London Math. Soc. 1 (1926), 98-99.
14. George Pólya, Über trigonometrische Integrale mit nur reellen Nullstellen, J. Reine Angew. Math. 158 (1927), 6-18.
15. George Pólya, Collected papers, The MIT Press, Cambridge, Mass.-London, 1974, Vol. II: Location of zeros, Edited by R. P. Boas, Mathematicians of Our Time, Vol. 8. MR 0505094 (58:21342)
16. Walter Rudin, Real and complex analysis, third ed., McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York, 1987. MR 0924157 (88k:00002)
17. David Ruelle, Statistical mechanics: Rigorous results, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0289084 (44:6279)
18. E. C. Titchmarsh, The theory of the Riemann zeta-function, second ed., Clarendon Press Oxford University Press, Oxford, 1986, Revised by D. R. Heath-Brown. MR 0882550 (88c:11049)