Abstract
Let \(\mathcal{F}\) be either the set of all bounded holomorphic functions or the set of all m-homogeneous polynomials on the unit ball of ℓr. We give a systematic study of the sets of all u ∊ ℓr for which the monomial expansion \(\sum\nolimits_\alpha {{{{\partial ^\alpha }f(0)} \over {\alpha !}}{u^\alpha }} \) of every f ∊ \(\mathcal{F}\) converges. Inspired by recent results from the general theory of Dirichlet series, we establish as our main tool independently interesting, upper estimates for the unconditional basis constants of spaces of polynomials on ℓr spanned by finite sets of monomials.
Similar content being viewed by others
References
R. M. Aron and J. Globevnik, Analytic functions on c 0, Rev. Mat. Complut. 2 (1989), 27–33.
R. Balasubramanian, B. Calado and H. Queffélec, The Bohr inequality for ordinary Dirichlet series, Studia Math. 175 (2006), 285–304.
M. Balazard, Remarques sur un théorème de G. Hálasz et A. Sárközy, Bull. Soc. Math. France 117 (1989), 389–413.
F. Bayart, Maximum modulus of random polynomials, Q. J. Math. 63 (2010), 21–39.
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. 368 (2017), 837–876.
F. Bayart, D. Pellegrino and J. B. Seoane-Sepúlveda, The Bohr radius of the n-dimensional polydisk is equivalent to \(\sqrt {({\rm{log}}\;n)/n} \), Adv. Math. 264 (2014), 726–746.
H. P. Boas, Majorant series, J. Korean Math. Soc. 37 (2000), 321–337.
H. Bohr, Über die Bedeutung der Potenzreihen unendlich vieler Variabeln in der Theorie der Dirichletschen Reihen ∑ \({{{a_n}} \over {{n^2}}}\), Nachr. Ges. Wiss. Göttingen, Math. Phys. Kl. (1913), 441–488.
R. de la Bretèche, Sur l’ordre de grandeur des polynômes de Dirichlet, Acta Arith. 134 (2008), 141–148.
O. F. Brevig, On the Sidon constant for Dirichlet polynomials, Bull. Sci. Math. 138 (2014), 656–664.
A. Defant and L. Frerick, The Bohr radius of the unit ball of ℓ p n, J. Reine Angew. Math. 2011 (2011), 131–147.
A. Defant, L. Frerick, J. Ortega-Cerdà, M. Ounaïes and K. Seip, The Bohnenblust-Hille inequality for homogeneous polynomials is hypercontractive, Ann. of Math. 174 (2011), 485–497.
A. Defant, D. García and M. Maestre, Bohr power series theorem and local Banach space theory, J. Reine Angew. Math. 557 (2003), 173–197.
A. Defant and N. Kalton, Unconditionality in spaces of m-homogeneous polynomials, Q. J. Math. 56 (2005), 53–64.
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.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford University Press, Oxford, 1938.
J.-P. Kahane, Some Random Series of Functions, Cambridge Stud. Adv. Math., Cambridge University Press, Cambridge, 1993.
S. V. Konyagin and H. Queffelec, The Translation ½ in the Theory of Dirichlet Series, Real Anal. Exchange 27 (2001), 155–176.
L. Lempert, The Dolbeault complex in infinite dimensions II, J. Amer. Math. Soc. 12 (1999), 775–793.
H. Queffélec and M. Queffélec, Diophantine Approximation and Dirichlet Series, Hindustan Book Agency, New Delhi, 2013.
R. A. Ryan, Holomorphic mappings on ℓ 1, Trans. Amer. Math. Soc. 302 (1987), 797.
Acknowledgment
The authors thank the referee for the careful reading of the manuscript and remarks regarding Section 3.3.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bayart, F., Defant, A. & Schlüters, S. Monomial convergence for holomorphic functions on ℓr. JAMA 138, 107–134 (2019). https://doi.org/10.1007/s11854-019-0022-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11854-019-0022-x