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Fourier transforms having only real zeros
Author:
David A. Cardon
Journal:
Proc. Amer. Math. Soc. 133 (2005), 1349-1356
MSC (2000):
Primary 42A38, 30C15
Posted:
October 18, 2004
MathSciNet review:
2111941
Full-text PDF Free Access
Abstract |
References |
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Additional Information
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  has only real zeros.
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- 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)
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- David A. Cardon, Convolution operators and zeros of entire functions, Proc. Amer. Math. Soc. 130 (2002), no. 6, 1725-1734. MR 1887020 (2002m:30006)
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- 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
 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 -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)
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Additional Information
David A. Cardon
Affiliation:
Department of Mathematics, Brigham Young University, Provo, Utah 84602
Email:
cardon@math.byu.edu
DOI:
http://dx.doi.org/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
Article copyright:
© Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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