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Fourier transforms and the Hermite-Biehler theorem

Authors: G. Csordas and R. S. Varga
Journal: Proc. Amer. Math. Soc. 107 (1989), 645-652
MSC: Primary 30C15; Secondary 30D15, 33A70, 42A38
MathSciNet review: 982401
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Abstract: A new necessary and sufficient condition for real entire functions, represented by Fourier transforms, to have only real zeros is proved. An application of this result to the Riemann $ \xi $-function is also given.

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  • [AK] N. I. Ahiezer and M. G. Krein, Some questions in the theory of moments, Transl. Math. Mono., vol. 2, Amer. Math. Soc., Providence, R.I., 1962. MR 0167806 (29:5073)
  • [B] N. G. de Bruijn, The roots of trigonometric integrals, Duke Math. J. 17 (1950), 197-226. MR 0037351 (12:250a)
  • [CNV] G. Csordas, T. S. Norfold and R. S. Varga, The Riemann Hypothesis and the Turán inequalities, Trans. Amer. Math. Soc. 296 (1986), 521-541. MR 846596 (87i:11109)
  • [CV1] G. Csordas and R. S. Varga, Moment inequalities and the Riemann Hypothesis, Constr. Approx. 4 (1988), 175-198. MR 932653 (89f:30010)
  • [CV2] -, Integral transforms and the Laguerre-Pólya class, Complex Analysis, (to appear).
  • [CV3] -, Necessary and sufficient conditions and the Riemann Hypothesis, (to appear).
  • [J] J. L. W. V. Jensen, Recherches sur la théorie des équations, Acta Math. 36 (1913), 181-195. MR 1555086
  • [L] E. Laguerre, Oeuvres, vol. I, Gauthier-Villars, Paris, 1898.
  • [Le] B. Ja. Levin, Distribution of zeros of entire functions, Transl. Math. Mono., vol. 5, Amer. Math. Soc., Providence, R.I., 1964; rev. ed., 1980. MR 589888 (81k:30011)
  • [M] N. N. Meiman, On the distribution of zeros of an integral function (Russian), Dokl. Akad. Nauk SSR 40 (1943), 179-181.
  • [P1] G. Pólya, Über trigonometrische Integrale mit nur reellen Nullstellen, J. Reine Angew. Math. 158 (1927), 6-18.
  • [P2] -, Über die algebraisch-funktionentheoritischen Untersuchungen von J. L. W. V. Jensen, Kgl. Danske Vid. Sel. Math.-Fys. Medd. 7 (1927), 3-33.
  • [P3] -, Collected papers, vol. II: Location of zeros, R. P. Boas, ed., MIT Press, Cambridge, 1974. MR 0505093 (58:21341)
  • [S] H. Skovgaard, On inequalities of the Turán type, Math. Scand. 2 (1954), 65-73. MR 0063415 (16:118d)
  • [T] E. C. Titchmarsh, The theory of the Riemann zeta function, second edition (revised by D. R. Heath-Brown), Oxford Science Publications, 1986. MR 882550 (88c:11049)

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Keywords: Fourier transforms, Laguerre-Pólya class, Laguerre inequalities, Hermite-Biehler theorem, Riemann Hypothesis, Riemann $ \xi $-function
Article copyright: © Copyright 1989 American Mathematical Society

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