Functiones et Approximatio Commentarii Mathematici

Bohr's strips for Dirichlet series in Banach spaces

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

Full-text: Open access

Abstract

Each Dirichlet series $D = \sum_{n=1}^{\infty} a_n \frac{1}{n^s}$, with variable $s \in \mathbb{C}$ and coefficients $a_n \in \mathbb{C}$, has a so called Bohr strip, the largest strip in $\mathbb{C}$ on which $D$ converges absolutely but not uniformly. The classical Bohr-Bohnenblust-Hille theorem states that the width of the largest possible Bohr strip equals $1/2$. Recently, this deep work of Bohr, Bohnenblust and Hille from the beginning of the last century was revisited by various authors. New methods from different fields of modern analysis (e.g. probability theory, number theory, functional and Fourier analysis) allow to improve the Bohr-Bohnenblust-Hille cycle of ideas, and to extend it to new settings, in particular to Dirichlet series which coefficients in Banach spaces. We survey on various aspects of these new developments.

Article information

Source
Funct. Approx. Comment. Math., Volume 44, Number 2 (2011), 165-189.

Dates
First available in Project Euclid: 22 June 2011

Permanent link to this document
https://projecteuclid.org/euclid.facm/1308749122

Digital Object Identifier
doi:10.7169/facm/1308749122

Mathematical Reviews number (MathSciNet)
MR2841177

Zentralblatt MATH identifier
1227.30003

Subjects
Primary: 32A05: Power series, series of functions
Secondary: 46B07: Local theory of Banach spaces 46B09: Probabilistic methods in Banach space theory [See also 60Bxx] 46G20: Infinite-dimensional holomorphy [See also 32-XX, 46E50, 46T25, 58B12, 58C10]

Keywords
Dirichlet series power series polynomials Banach spaces

Citation

Defant, Andreas; García, Domingo; Maestre, Manuel; Sevilla-Peris, Pablo. Bohr's strips for Dirichlet series in Banach spaces. Funct. Approx. Comment. Math. 44 (2011), no. 2, 165--189. doi:10.7169/facm/1308749122. https://projecteuclid.org/euclid.facm/1308749122


Export citation

References

  • R. Balasubramanian, B. Calado and H. Queffélec, The Bohr inequality for ordinary Dirichlet series, Studia Math. 175(3) (2006), 285–304.
  • G. Bennett, Inclusion mappings between $l\spp$ spaces, J. Functional Analysis 13 (1973), 20–27.
  • H. F. Bohnenblust and E. Hille, On the absolute convergence of Dirichlet series, Ann. of Math. (2) 32(3) (1931), 600–622.
  • H. Bohr, Bidrag til de Dirichlet'ske Rækkers Theori (Contributions to the theory of Dirichlet Series), PhD thesis, University of Copenhagen, 1910; Translated into English in Collected Mathematical Works. Vol. III. Dansk Matematisk Forening, København, 1952, Document S–1 (118 pages).
  • H. Bohr, Über die Bedeutung der Potenzreihen unendlich vieler Variabeln in der Theorie der Dirichlet–schen Reihen $\sum\,\fraca_nn^2$, Nachr. Ges. Wiss. Göttingen, Math. Phys. Kl., pages 441–488, 1913.
  • H. Bohr, Über die gleichmäßige Konvergenz Dirichletscher Reihen, J. Reine Angew. Math. 143 (1913), 203–211.
  • H. Bohr, A theorem concerning power series, Proc. London Math. Soc. (2) 13 (1914), 1–5.
  • P. J. Boland and S. Dineen, Fonctions holomorphes sur des espaces pleinement nucléaires, C. R. Acad. Sci. Paris Sér. A-B 286(25) (1978), A1235–A1237.
  • F. Bombal, D. Pérez-García, and I. Villanueva, Multilinear extensions of Grothendieck's theorem, Q. J. Math. 55(4) (2004), 441–450.
  • B. Carl, Absolut-$(p,\,1)$-summierende identische Operatoren von $l\sbu$ in $l\sbv$, Math. Nachr. 63 (1974), 353–360.
  • A. M. Davie, Quotient algebras of uniform algebras, J. London Math. Soc. (2) 7 (1973), 31–40.
  • R. de la Bretèche, Sur l'ordre de grandeur des polynômes de Dirichlet, Acta Arith. 134(2) (2008), 141–148.
  • A. Defant, J. C. Díaz, D. García, and M. Maestre, Unconditional basis and Gordon-Lewis constants for spaces of polynomials, J. Funct. Anal. 181(1) (2001), 119–145.
  • A. Defant and K. Floret, Tensor norms and operator ideals, volume 176 of North-Holland Mathematics Studies, North-Holland Publishing Co., Amsterdam, 1993.
  • A. Defant, L. Frerick, J. Ortega-Cerdà, M. Ounaïes, and K. Seip, The Bohnenblust–Hille inequality for homogeneous polynomials is hypercontractive, Annals of Mathematics, to appear, 2011.
  • A. Defant, D. García, and M. Maestre, Bohr's power series theorem and local Banach space theory, J. Reine Angew. Math. 557 (2003), 173–197.
  • 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, M. Maestre, and U. Schwarting, Bohr radii of vector valued holomorphic functions, preprint.
  • A. Defant, D. Popa, and U. Schwarting, Coordinatewise multiple summing operators in Banach spaces, J. Funct. Anal. 259(1) (2010), 220–242.
  • A. Defant and U. Schwarting, Queffélec numbers, in preparation.
  • A. Defant and U. Schwarting, Bohr's strips and radii – a macro- and a macroscopic view, Note di Mat., to appear, 2011.
  • A. Defant and P. Sevilla-Peris, A new multilinear insight on Littlewood's 4/3-inequality, J. Funct. Anal. 256(5) (2009), 1642–1664.
  • A. Defant and P. Sevilla-Peris, Convergence of Dirichlet polynomials in Banach spaces, Trans. Amer. Math. Soc. 363(2) (2011), 681–697.
  • A. Defant and P. Sevilla-Peris, Convergence of monomial expansions in Banach spaces, Q. J. Math., available on-line, 2011.
  • J. Diestel, H. Jarchow, and A. Tonge, Absolutely summing operators, volume 43 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1995.
  • S. Dineen, Complex analysis on infinite-dimensional spaces, Springer Monographs in Mathematics, Springer-Verlag London Ltd., London, 1999.
  • G. Hardy and M. Riesz, The general theory of Dirichlet's series, Cambridge University Press, Cambridge, 1915.
  • L. A. Harris, Bounds on the derivatives of holomorphic functions of vectors, In Analyse fonctionnelle et applications (Comptes Rendus Colloq. Analyse, Inst. Mat., Univ. Federal Rio de Janeiro, Rio de Janeiro, 1972), pages 145–163. Actualités Aci. Indust., No. 1367. Hermann, Paris, 1975.
  • D. Hilbert, Wesen und Ziele einer Analysis der unendlichvielen unabhängigen Variablen, Rend. del Circolo mat. di Palermo 27 (1909), 59–74.
  • S. Kaijser, Some results in the metric theory of tensor products, Studia Math. 63(2) (1978), 157–170.
  • S. V. Konyagin and H. Queffélec, The translation $\frac12$ in the theory of Dirichlet series, Real Anal. Exchange 27(1) (2001/02), 155–175.
  • S. Kwapień, Some remarks on $(p,\,q)$-absolutely summing operators in $l\sbp$-spaces, Studia Math. 29 (1968), 327–337.
  • J. E. Littlewood, On bounded bilinear forms in an infinite number of variables, Quarterly Journ. (Oxford Series) 1 (1930), 164–174.
  • B. Maurizi and H. Queffélec, Some remarks on the algebra of bounded Dirichlet series, J. Fourier Anal Appl, DOI 10.1007/s00041-009-9112-y, 2010.
  • P. Mellon, The polarisation constant for $\rm JB\sp \ast$-triples, Extracta Math. 9(3) (1994), 160–163.
  • H. Queffélec, H. Bohr's vision of ordinary Dirichlet series; old and new results, J. Anal. 3 (1995), 43–60.
  • J. Sawa, The best constant in the Khintchine inequality for complex Steinhaus variables, the case $p=1$, Studia Math. 81(1) (1985), 107–126.
  • M. Talagrand, Cotype and $(q,1)$-summing norm in a Banach space, Invent. Math. 110(3) (1992), 545–556.
  • O. Toeplitz, Über eine bei den Dirichletschen Reihen auftretende Aufgabe aus der Theorie der Potenzreihen von unendlichvielen Veränderlichen, Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, pages 417–432, 1913.
  • N. Tomczak-Jaegermann, Banach-Mazur distances and finite-dimensional operator ideals, volume 38 of Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientific & Technical, Harlow, 1989.
  • A. Tonge, Polarization and the two-dimensional Grothendieck inequality, Math. Proc. Cambridge Philos. Soc. 95(2) (1984), 313–318.