Proceedings of the American Mathematical Society

Author(s): David A. Cardon
Journal: Proc. Amer. Math. Soc. 133 (2005), 1349-1356.
MSC (2000): Primary 42A38, 30C15
Posted: October 18, 2004
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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)

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 42A38, 30C15

David A. Cardon
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602
Email: cardon@math.byu.edu

DOI: 10.1090/S0002-9939-04-07677-4
PII: S 0002-9939(04)07677-4
Keywords: Fourier transform, zeros of entire functions, Laguerre-P\'olya class
Received by editor(s): September 23, 2003
Received by editor(s) in revised form: December 23, 2003
Posted: October 18, 2004
Communicated by: Joseph A. Ball
Copyright of article: Copyright 2004, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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