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MAXIMUM MODULI OF UNIMODULAR POLYNOMIALS

  • Published : 2004.01.01

Abstract

Let $\Sigma_{$\mid$\alpha$\mid$=m}\;s_{\alpha}z^{\alpha},\;z\;{\in}\;{\mathbb{C}}^n$ be a unimodular m-homogeneous polynomial in n variables (i.e. $$\mid$s_{\alpha}$\mid$\;=\;1$ for all multi indices $\alpha$), and let $R\;{\subset}\;{\mathbb{C}}^n$ be a (bounded complete) Reinhardt domain. We give lower bounds for the maximum modules $sup_{z\;{\in}\;R\;$\mid$\Sigma_{$\mid$\alpha$\mid$=m}\;s_{\alpha}z^{\alpha}$\mid$$, and upper estimates for the average of these maximum moduli taken over all possible m-homogeneous Bernoulli polynomials (i.e. $s_{\alpha}\;=\;{\pm}1$ for all multi indices $\alpha$). Examples show that for a fixed degree m our estimates, for rather large classes of domains R, are asymptotically optimal in the dimension n.

Keywords

References

  1. J. Korean Math. Soc. v.37 no.2 Majorant series, Several complex variables H.Boas
  2. Notices Amer. Math. Soc. v.44 no.11 The football player and the infinite series H.Boas
  3. Proc. Amer. Math. Soc. v.125 no.10 Bohr's power series theorem in several variables H.P.Boas;D.Khavinson https://doi.org/10.1090/S0002-9939-97-04270-6
  4. Proc. London Math. Soc. v.13 no.2 A theorem concerning power series H.Bohr
  5. Mathematisch-Physikalische Klasse Uber die Bedeutung der Potenzreihen unendlich vieler Variabeln in der Theorie der Dirichletschen Reihen $\Sigma{a}_{n}/{n}^{8}$, Nachrichten von der Koniglichen Gesellschaft der Wissenschafen zu Gottingen H.Bohr
  6. Ann. of Math. v.32 On the absolute convergence of Dirichlet series H.F.Bohnenblust;E.Hille https://doi.org/10.2307/1968255
  7. Application aux produits d'espaces de Wiener abstraits, Seminaire Maurey-Schwartz no.Exp. 19 Series de variable aleatories Gaussiens a valeur dons EⓧF S.Chevet
  8. J. Funct. Anal. v.181 Unconditional basis and Gordon-Lewis constants for spaces of polynomials A.Defant;J.C.Diaz;D.Garcia;M.Maestre https://doi.org/10.1006/jfan.2000.3702
  9. J. reine angew. Math. v.557 Bohr's power series theorem and local Banach space theory A.Defant;D.Garcia;M.Maestre
  10. North-Holland Math. Stud. v.189 Summing inclusion maps between symmetric sequence spaces, a survey, Recent progress in functional analysis A.Defant;M.Mastylo;C.Michels
  11. North-Holland Math. Studies v.176 Tensor Norms and Operator Ideals A.Defant;K.Floret https://doi.org/10.1016/S0304-0208(01)80035-9
  12. Springer Monographs in mathematics Complex Analysis on Infinite Dimensional Banach Spaces S.Dineen
  13. Studia Math. v.94 Absolute bases, tensor products and a theorem of Bohr S.Dineen;R.M.Timoney https://doi.org/10.4064/sm-94-3-227-234
  14. Bull. Soc. Rou. Sci. Liego v.60 no.6 On a problem of H. Bohr S.Dineen;R.M.Timoney
  15. C. R. Acad. Sci. Paris, Ser. A-B v.278 Des resultats nouveaux sur les processus gaussiens X.Fernique
  16. Note Mat. v.17 Natural norms on symmetric tensor products of normed spaces K.Floret
  17. Israel J. Math. v.50 no.4 Some inequalities for Gaussians processes and applications Y.Gordon https://doi.org/10.1007/BF02759761
  18. Cambridge Stud. Adv. Math. v.5 Some random series of functions, second ed. J.P.Kahane
  19. Jour. Math. Anal. Appl. v.226 Type and cotype of Musielak-Orlicz sequence spaces E.Katirtzoglou https://doi.org/10.1006/jmaa.1998.6089
  20. Studia Math. v.95 Combinatorial and probabilistic inequalities and their applications to Banach space theory Ⅱ S.Kwapien;C.Schutt https://doi.org/10.4064/sm-95-2-141-154
  21. Isoperimetry and processes Probability in Banach spaces M.Ledoux;M.Talagrand
  22. Classical Banach spaces Ⅰ and Ⅱ J.Lindenstrauss;L.Tzafriri
  23. Studia Math. v.68 The Schur multiplication in tensor algebras A.M.Mantero;A.Tonge https://doi.org/10.4064/sm-68-1-1-24
  24. Proc. London Math. Soc. v.38 Banach algebras and von Neumann's inequality A.M.Mantero;A.Tonge https://doi.org/10.1112/plms/s3-38.2.309
  25. Bell Syst. Tech. J. v.41 The one-sided barrier problem for Gausssian noise D.Slepian https://doi.org/10.1002/j.1538-7305.1962.tb02419.x
  26. Israel J. Math v.30 The projection constant of finite-dimesional spaces whose unconditional basis constant is 1 C.Schutt https://doi.org/10.1007/BF02761071
  27. Compositio Math. v.40 On nearly Euclidean decompositions for some classes of Banach spaces S.J.Szarek;N.Tomczak-Jaegermann
  28. Longman Scienticic & Technical Banach-Mazur Distances and Finite-Dimensional Operators Ideals N.Tomczak-Jaegermann

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