Two complete and minimal systems associated with the zeros of the Riemann zeta function

Burnol, Jean-François

Journal de théorie des nombres de Bordeaux, Tome 16 (2004) no. 1, pp. 65-94.

Nous relions trois thèmes restés jusqu’alors distincts : les propriétés hilbertiennes des zéros de Riemann, la “formule duale de Poisson” de Duffin-Weinberger (que nous appelons formule de co-Poisson), les espaces de fonctions entières “de Sonine” définis et étudiés par de Branges. Nous déterminons dans quels espaces de Sonine (étendus) les zéros forment un système complet, ou minimal. Nous obtenons des résultats généraux concernant la distribution des zéros des fonctions entières de de Branges-Sonine. Nous attirons l’attention sur certaines distributions liées à la transformation de Fourier et qui sont apparues dans nos travaux antérieurs.

We link together three themes which had remained separated so far: the Hilbert space properties of the Riemann zeros, the “dual Poisson formula” of Duffin-Weinberger (also named by us co-Poisson formula), and the “Sonine spaces” of entire functions defined and studied by de Branges. We determine in which (extended) Sonine spaces the zeros define a complete, or minimal, system. We obtain some general results dealing with the distribution of the zeros of the de-Branges-Sonine entire functions. We draw attention onto some distributions associated with the Fourier transform and which we introduced in our earlier works.

[1] R. P. Boas, Sums representing Fourier transforms, Proc. Am. Math. Soc. 3 (1952), 444–447. | MR | Zbl

[2] L. de Branges, Self-reciprocal functions, J. Math. Anal. Appl. 9 (1964) 433–457. | MR | Zbl

[3] L. de Branges, Hilbert spaces of entire functions, Prentice Hall Inc., Englewood Cliffs, 1968. | MR | Zbl

[4] L. de Branges, The convergence of Euler products, J. Funct. Anal. 107 (1992), no. 1, 122–210. | MR | Zbl

[5] L. de Branges, A conjecture which implies the Riemann hypothesis, J. Funct. Anal. 121 (1994), no. 1, 117–184. | MR | Zbl

[6] J.-F. Burnol, Sur certains espaces de Hilbert de fonctions entières, liés à la transformation de Fourier et aux fonctions L de Dirichlet et de Riemann, C. R. Acad. Sci. Paris, Ser. I 333 (2001), 201–206. | MR | Zbl

[7] J.-F. Burnol, On Fourier and Zeta(s), 50 p., Habilitationsschrift (2001-2002), Forum Mathematicum, to appear (2004). | MR | Zbl

[8] J.-F. Burnol, Sur les “espaces de Sonine” associés par de Branges à la transformation de Fourier, C. R. Acad. Sci. Paris, Ser. I 335 (2002), 689–692. | MR | Zbl

[9] J.-F. Burnol, Des équations de Dirac et de Schrödinger pour la transformation de Fourier, C. R. Acad. Sci. Paris, Ser. I 336 (2003), 919–924. | MR | Zbl

[10] R. J. Duffin, Representation of Fourier integrals as sums I, Bull. Am. Math. Soc. 51 (1945), 447–455. | MR | Zbl

[11] R. J. Duffin, Representation of Fourier integrals as sums II, Proc. Am. Math. Soc. 1 (1950), 250–255. | MR | Zbl

[12] R. J. Duffin, Representation of Fourier integrals as sums III, Proc. Am. Math. Soc. 8 (1957), 272–277. | MR | Zbl

[13] R. J. Duffin, H. F. Weinberger, Dualizing the Poisson summation formula, Proc. Natl. Acad. Sci. USA 88 (1991), 7348–7350. | MR | Zbl

[14] R. J. Duffin, H. F. Weinberger, On dualizing a multivariable Poisson summation formula, Journ. of Fourier Anal. and Appl. 3 (5) (1997), 487–497. | MR | Zbl

[15] H. Dym, H.P. McKean, Fourier series and integrals, Academic Press, 1972. | MR | Zbl

[16] H. Dym, H.P. McKean, Gaussian processes, function theory, and the inverse spectral problem, Probability and Mathematical Statistics, Vol. 31. Academic Press, New York-London, 1976. | MR | Zbl

[17] M. L. Gorbachuk, V. I. Gorbachuk, M. G. Krein’s lectures on entire operators, Operator Theory: Advances and Applications, 97. Birkhäuser Verlag, Basel, 1997. | Zbl

[18] K. Hoffman, Banach spaces of analytic functions, Reprint of the 1962 original. Dover Publications, Inc., New York, 1988. | MR | Zbl

[19] M.G. Krein, Theory of entire functions of exponential type (in Russian), Izv. Akad. Nauk. SSSR, Ser. Mat. 11 (1947), No. 4, 309–326. | MR | Zbl

[20] B.Y. Levin, Distribution of Zeros of Entire Functions, American Mathematical Society, Providence 1980. Transl. and rev. from the 1956 Russian and 1962 German editions. | MR | Zbl

[21] R.E.A.C. Paley, N. Wiener, Fourier Transforms in the Complex Domain, Amer. Math. Soc., Providence, Rhode Island, 1934. | MR | Zbl

[22] J. Rovnyak, V. Rovnyak, Sonine spaces of entire functions, J. Math. Anal. Appl., 27 (1969), 68–100. | MR | Zbl

[23] N. Sonine, Recherches sur les fonctions cylindriques et le développement des fonctions continues en séries, Math. Ann. 16 (1880), 1–80. | MR

[24] E. C. Titchmarsh, The Theory of the Riemann-Zeta Function, 2nd ed. Edited and with a preface by D. R. Heath-Brown. Clarendon Press, Oxford 1986. | MR | Zbl

[25] H. F. Weinberger, Fourier transforms of Moebius series. Dissertation (1950), Carnegie-Mellon University, Pittsburgh.