Sums of entire functions having only real zeros
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- by Steven R. Adams and David A. Cardon PDF
- Proc. Amer. Math. Soc. 135 (2007), 3857-3866 Request permission
Abstract:
We show that certain sums of products of Hermite-Biehler entire functions have only real zeros, extending results of Cardon. As applications of this theorem, we construct sums of exponential functions having only real zeros, we construct polynomials having zeros only on the unit circle, and we obtain the three-term recurrence relation for an arbitrary family of real orthogonal polynomials. We discuss a similarity of this result with the Lee-Yang Circle Theorem from statistical mechanics. Also, we state several open problems.References
- David A. Cardon, Convolution operators and zeros of entire functions, Proc. Amer. Math. Soc. 130 (2002), no. 6, 1725–1734. MR 1887020, DOI 10.1090/S0002-9939-01-06351-1
- David A. Cardon, Sums of exponential functions having only real zeros, Manuscripta Math. 113 (2004), no. 3, 307–317. MR 2129307, DOI 10.1007/s00229-003-0414-0
- David A. Cardon, Fourier transforms having only real zeros, Proc. Amer. Math. Soc. 133 (2005), no. 5, 1349–1356. MR 2111941, DOI 10.1090/S0002-9939-04-07677-4
- 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
- T. D. Lee and C. N. Yang, Statistical theory of equations of state and phase transitions. II. Lattice gas and Ising model, Phys. Rev. (2) 87 (1952), 410–419. MR 53029, DOI 10.1103/PhysRev.87.410
- B. Ja. Levin, Distribution of zeros of entire functions, Revised edition, Translations of Mathematical Monographs, vol. 5, American Mathematical Society, Providence, R.I., 1980. Translated from the Russian by R. P. Boas, J. M. Danskin, F. M. Goodspeed, J. Korevaar, A. L. Shields and H. P. Thielman. MR 589888
- George Pólya, Bemerkung über die Integraldarstellung der Riemannsche $\xi$-Funktion, Acta Math. 48 (1926), 305–317.
- David Ruelle, Statistical mechanics, World Scientific Publishing Co., Inc., River Edge, NJ; Imperial College Press, London, 1999. Rigorous results; Reprint of the 1989 edition. MR 1747792, DOI 10.1142/4090
- Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975. MR 0372517
Additional Information
- Steven R. Adams
- Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602
- David A. Cardon
- Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602
- Email: cardon@math.byu.edu
- Received by editor(s): August 9, 2006
- Published electronically: August 29, 2007
- Communicated by: Juha M. Heinonen
- © Copyright 2007 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 135 (2007), 3857-3866
- MSC (2000): Primary 30C15; Secondary 30D05
- DOI: https://doi.org/10.1090/S0002-9939-07-09103-4
- MathSciNet review: 2341936