Functiones et Approximatio Commentarii Mathematici

Dirichlet series from the infinite dimensional point of view

Andreas Defant, Domingo García, Manuel Maestre, and Pablo Sevilla-Peris

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Abstract

A classical result of Harald Bohr linked the study of convergent and bounded Dirichlet series on the right half plane with bounded holomorphic functions on the open unit ball of the space $c_0$ of complex null sequences. Our aim here is to show that many questions in Dirichlet series have very natural solutions when, following Bohr's idea, we translate these to the infinite dimensional setting. Some are new proofs and other new results obtained by using that point of view.

Article information

Source
Funct. Approx. Comment. Math., Volume 59, Number 2 (2018), 285-304.

Dates
First available in Project Euclid: 26 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.facm/1529978435

Digital Object Identifier
doi:10.7169/facm/1741

Mathematical Reviews number (MathSciNet)
MR3892307

Zentralblatt MATH identifier
07055557

Subjects
Primary: 30B50: Dirichlet series and other series expansions, exponential series [See also 11M41, 42-XX]
Secondary: 46G20: Infinite-dimensional holomorphy [See also 32-XX, 46E50, 46T25, 58B12, 58C10]

Keywords
Dirichlet series Bohr transform holomorphic function Banach space

Citation

Defant, Andreas; García, Domingo; Maestre, Manuel; Sevilla-Peris, Pablo. Dirichlet series from the infinite dimensional point of view. Funct. Approx. Comment. Math. 59 (2018), no. 2, 285--304. doi:10.7169/facm/1741. https://projecteuclid.org/euclid.facm/1529978435


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References

  • R.,M. Aron and P.,D. Berner, A Hahn-Banach extension theorem for analytic mappings, Bull. Soc. Math. France 106 (1978), 3–24.
  • R.,M. Aron, F. Bayart, P. Gauthier, M. Maestre and V. Nestoridis, Dirichlet approximation and universal Dirichlet series, Proc. Amer. Math. Soc. 145 (2017), no. 10, 4449–4464.
  • R.,M. Aron, B.,J. Cole and T.,W. Gamelin, Weak-star continuous analytic functions, Canad. J. Math. 47 (1995), no. 4, 673–683.
  • A. Aleman, J.-F. Olsen and E. Saksman, Fourier multipliers for Hardy spaces of Dirichlet series, Int. Math. Res. Not. IMRN 16 (2014), 4368–4378.
  • R. Balasubramanian, B. Calado and H. Queffélec, The Bohr inequality for ordinary Dirichlet series, Studia Math. 175(3) (2006), 285–304.
  • F. Bayart, The product of two Dirichlet series, Acta Arith. 111 (2004), no. 2, 141–152.
  • F. Bayart, Compact composition operators on a Hilbert space of Dirichlet series, Illinois J. Math. 47 (2003), no. 3, 725–743.
  • F. Bayart, Hardy spaces of Dirichlet series and their composition operators, Monatsh. Math. 136(3) (2002), 203–236.
  • F. Bayart, A. Defant, L. Frerick, M. Maestre and P. Sevilla-Peris, Multipliers of Dirichlet series and monomial series expansions of holomorphic functions in infinitely many variables, Math. Ann. 837 (2017), 837–876.
  • F. Bayart, C. Finet, D. Li and H. Queffélec, Composition operators on the Wiener-Dirichlet algebra, J. Operator Theory 60 (2008), no. 1, 45–70.
  • F. Bayart, S.,V. Konyagin and H. Queffélec, Convergence almost everywhere and divergence everywhere of Taylor and Dirichlet series, Real Anal. Exchange 29 (2003/04), no. 2, 557–586.
  • F. Bayart and A. Mouze, Sur l'irréductibilité dans l'anneau des séries de Dirichlet analytiques, Publ. Mat. 49 (2005), no. 1, 93–110.
  • F. Bayart and A. Mouze, Division et composition dans l'anneau des séries de Dirichlet analytiques, Ann. Inst. Fourier 53 (2003), no. 7, 2039–2060.
  • F. Bayart, H. Queffélec and K. Seip, Approximation numbers of composition operators on $\mathcal{H}^p$ spaces of Dirichlet series, Ann. Inst. Fourier (Grenoble) 66 (2016), no. 2, 551–588.
  • H.,F. Bohnenblust and E. Hille, On the absolute convergence of Dirichlet series, Ann. of Math. (2) 32(3) (1931), 600–622.
  • A. Bondarenko, W. Heap and K. Seip, An inequality of Hardy-Littlewood type for Dirichlet polynomials, J. Number Theory 150 (2015), 191–205.
  • R. de la Bretèche, Sur l'ordre de grandeur des polynômes de Dirichlet, Acta Arith. 134(2) (2008), 141–148.
  • D. Carando, A. Defant and P. Sevilla-Peris, Almost sure-sign convergence of Hardy-type Dirichlet series, J. Anal. Math., to appear.
  • D. Carando, A. Defant and P. Sevilla-Peris, The Bohnenblust–Hille inequality combined with an inequality of Helson, Proc. Amer. Math. Soc., 143 (2015), no. 12, 5233–5238.
  • D. Carando, A. Defant and P. Sevilla-Peris, Bohr's absolute convergence problem for ${\mathcal H}_p$-Dirichlet series in Banach spaces, Anal. PDE 7(2) (2014), 513–527.
  • J. Castillo-Medina, D. García and M. Maestre, Isometries between spaces of multiple Dirichlet series, preprint.
  • A.,M. Davie and T. W. Gamelin, A theorem on polynomial-star approximation, Proc. Amer. Math. Soc. 106 (1989), 351-356.
  • A. Defant, L. Frerick, J. Ortega-Cerdà, M. Ounaïes and K. Seip, The Bohnenblust-Hille inequality for homogeneous polynomials is hypercontractive, Ann. of Math. (2) 174(1) (2011), 485–497.
  • A. Defant, D. García, M. Maestre and D. Pérez-García, Bohr's strip for vector valued Dirichlet series, Math. Ann. 342(3) (2008), 533–555.
  • A. Defant, D. García, M. Maestre and P. Sevilla-Peris, Dirichlet series and holomorphic functions in high dimensions, to appear in New Mathematical Monographs Series, Cambridge University Press.
  • A. Defant, M. Maestre and C. Prengel, Domains of convergence for monomial expansions of holomorphic functions in infinitely many variables, J. Reine Angew. Math. 634 (2009), 13–49.
  • T.,W. Gamelin, Uniform algebras, 2nd edition, Chelsea Press, 1984.
  • P.,M. Gauthier, R. Grothmann and W. Hengartner, Asymptotic maximum principles for subharmonic and plurisubharmonic functions, Canad. J. Math. 40 (1988), no. 2, 477–486.
  • H. Hedenmalm, Dirichlet series and functional analysis, in The legacy of Niels Henrik Abel, pages 673–684. Springer, Berlin, 2004.
  • H. Hedenmalm, P. Lindqvist and K. Seip, A Hilbert space of Dirichlet series and systems of dilated functions in $L^2(0,1)$, Duke Math. J. 86(1) (1997), 1–37.
  • H. Hedenmalm and E. Saksman, Carleson's convergence theorem for Dirichlet series, Pacific J. Math. 208 (2003), no. 1, 85–109.
  • H. Helson, Dirichlet series, Henry Helson, Berkeley, CA, 2005.
  • H. Helson, Hankel forms and sums of random variables, Studia Math. 176(1) (2006), 85–92.
  • H. Helson and D. Lowdenslager, Prediction theory and Fourier series in several variables, Acta Math. 99 (1958), 165–202.
  • G.,M. Henkin, Non-isomorphism of certain spaces of functions of different number of variables, Funt. Anal. Prilož 1 (1967), no. 4, 57–68.
  • D. Hilbert, Wesen und Ziele einer Analysis der unendlichvielen unabhängigen Variablen, Rend. Circolo Mat. Palermo, 27 (1909), 59–74.
  • S.,V. Konyagin and H. Queffélec, The translation $\frac12$ in the theory of Dirichlet series, Real Anal. Exchange 27(1) (2001/02), 155–175.
  • J.,F. Olsen and E. Saksman, On the boundary behaviour of the Hardy spaces of Dirichlet series and a frame bound estimate, J. Reine Angew. Math. 663 (2012), 33–66.
  • H. Queffélec, H. Bohr's vision of ordinary Dirichlet series; old and new results, J. Anal. 3 (1995), 43–60.
  • H. Queffélec and M. Queffélec, Diophantine approximation and Dirichlet series, HRI Lecture Notes Series, New Delhi, 2013.
  • H. Queffélec and K. Seip, Approximation numbers of composition operators on the $\mathcal{H}^2$ space of Dirichlet series, J. Funct. Anal. 268 (2015), no. 6, 1612–1648.
  • E. Saksman, and K. Seip, Integral means and boundary limits of Dirichlet series, Bull. Lond. Math. Soc. 41 (2009), no. 3, 411–422.
  • K. Seip, Zeros of functions in Hilbert spaces of Dirichlet series, Math. Z. 274 (2013), no. 3-4, 1327–1339.