Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Fourier transforms with only real zeros


Author: Charles M. Newman
Journal: Proc. Amer. Math. Soc. 61 (1976), 245-251
MSC: Primary 10H05; Secondary 42A68
MathSciNet review: 0434982
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The class of even, nonnegative, finite measures $ \rho $ on the real line such that for any $ b > 0$ the Fourier transform of $ \exp ( - b{t^2})d\rho (t)$ has only real zeros is completely determined. This result is then applied to the Riemann hypothesis.


References [Enhancements On Off] (What's this?)

  • [B] N. G. de Bruijn, The roots of trigonometric integrals, Duke Math. J. 17 (1950), 197–226. MR 0037351
  • [M] Morris Marden, The Geometry of the Zeros of a Polynomial in a Complex Variable, Mathematical Surveys, No. 3, American Mathematical Society, New York, N. Y., 1949. MR 0031114
  • [N1] Charles M. Newman, Zeros of the partition function for generalized Ising systems, Comm. Pure Appl. Math. 27 (1974), 143–159. MR 0484184
  • [N2] Charles M. Newman, Inequalities for Ising models and field theories which obey the Lee-Yang theorem, Comm. Math. Phys. 41 (1975), 1–9. MR 0376061
  • [N3] Charles M. Newman, Classifying general Ising models, Les méthodes mathématiques de la théorie quantique des champs (Colloq. Internat. CNRS, No. 248, Marseille, 1975) Éditions Centre Nat. Recherche Sci., Paris, 1976, pp. 273–288 (English, with French summary). MR 0479247
  • [P] George Pólya, Collected papers, The MIT Press, Cambridge, Mass.-London, 1974. Vol. 1: Singularities of analytic functions; Edited by R. P. Boas; Mathematicians of Our Time, Vol. 7. MR 0505093
  • [SG] Barry Simon and Robert B. Griffiths, The (𝜑⁴)₂ field theory as a classical Ising model, Comm. Math. Phys. 33 (1973), 145–164. MR 0428998
  • [T] E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, Oxford, at the Clarendon Press, 1951. MR 0046485

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 10H05, 42A68

Retrieve articles in all journals with MSC: 10H05, 42A68


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0434982-5
Article copyright: © Copyright 1976 American Mathematical Society