Convolution operators and zeros of entire functions

Cardon, David

doi:10.1090/S0002-9939-01-06351-1

Convolution operators and zeros of entire functions

Author: David A. Cardon
Journal: Proc. Amer. Math. Soc. 130 (2002), 1725-1734
MSC (2000): Primary 44A35, 30C15
DOI: https://doi.org/10.1090/S0002-9939-01-06351-1
Published electronically: October 17, 2001
MathSciNet review: 1887020
Full-text PDF Free Access

Abstract | References | Similar Articles | 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 convolution $(G*dF)(z)=\int_{-\infty}^{\infty} G(z-is)\,dF(s)$ has only real zeros.

1. Lars V. Ahlfors, Complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1978. An introduction to the theory of analytic functions of one complex variable; International Series in Pure and Applied Mathematics. MR 510197
2. Daniel Bump, Kwok-Kwong Choi, Pär Kurlberg, and Jeffrey Vaaler, A local Riemann hypothesis. I, Math. Z. 233 (2000), no. 1, 1–19. MR 1738342, https://doi.org/10.1007/PL00004786
3. N.B. de Bruijn, The Roots of Trigonometric Integrals, Duke Math. J. 17 (1950), 197-226. MR 12:250a
4. Thomas Craven and George Csordas, Differential operators of infinite order and the distribution of zeros of entire functions, J. Math. Anal. Appl. 186 (1994), no. 3, 799–820. MR 1293856, https://doi.org/10.1006/jmaa.1994.1334
5. George Csordas and Richard S. Varga, Integral transforms and the Laguerre-Pólya class, Complex Variables Theory Appl. 12 (1989), no. 1-4, 211–230. MR 1040921, https://doi.org/10.1080/17476938908814366
6. G. Csordas and R. S. Varga, Fourier transforms and the Hermite-Biehler theorem, Proc. Amer. Math. Soc. 107 (1989), no. 3, 645–652. MR 982401, https://doi.org/10.1090/S0002-9939-1989-0982401-1
7. George Csordas, Wayne Smith, and Richard S. Varga, Level sets of real entire functions and the Laguerre inequalities, Analysis 12 (1992), no. 3-4, 377–402. MR 1182637, https://doi.org/10.1524/anly.1992.12.34.377
8. George Csordas, Richard S. Varga, and István Vincze, Jensen polynomials with applications to the Riemann 𝜁-function, J. Math. Anal. Appl. 153 (1990), no. 1, 112–135. MR 1080122, https://doi.org/10.1016/0022-247X(90)90269-L
9. M.L. Eaton, A probability inequality for linear combinations of bounded random variables, Ann. Stat. 2 (1974), 609-614.
10. Martin Eisen, Introduction to mathematical probability theory, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1969. MR 0258078
11. Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, Francesco G. Tricomi, Higher transcendental functions, Vol. II. Based, in part, on notes left by Harry Bateman, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953. MR 15:419i
12. Bert Fristedt and Lawrence Gray, A modern approach to probability theory, Probability and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1997. MR 1422917
13. Dennis A. Hejhal, On a result of G. Pólya concerning the Riemann 𝜉-function, J. Analyse Math. 55 (1990), 59–95. MR 1094712, https://doi.org/10.1007/BF02789198
14. B. Ja. Levin, Distribution of zeros of entire functions, American Mathematical Society, Providence, R.I., 1964. MR 0156975
15. Michel Loève, Probability theory, Third edition, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1963. MR 0203748
16. Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157
17. Iosif Pinelis, Extremal probabilistic problems and Hotelling’s 𝑇² test under a symmetry condition, Ann. Statist. 22 (1994), no. 1, 357–368. MR 1272088, https://doi.org/10.1214/aos/1176325373
18. George Pólya, Bemerkung über die Integraldarstellung der Riemannsche $\xi$ -Funktion, Acta Math. 48 (1926), 305-317.
19. George Pólya, Über trigonometrische Integrale mit nur reellen Nullstellen, J. Reine Angew. Math. 158 (1927), 6-18.
20. G. Pólya and J. Schur, Über zwei Arten von Faktorfolgen in der Theorie der algebraischen Gleichungen, J. Reine Angew. Math. 144 (1914), 89-113.
21. Gabor Szegö, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., Vol. XXIII, 1939. MR 1:14b
22. E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1986. Edited and with a preface by D. R. Heath-Brown. MR 882550