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A complete characterization of coefficients of a.e. convergent orthogonal series and majorizing measures

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Abstract

We characterize sequences of numbers (a n ) such that ∑n≥1 a n Φ n converges a.e. for any orthonormal system (Φ n ) in any L 2-space. In our criteria we use the set A={∑m≥n||a m |2;n≥1}, majorizing measures on A, and new descriptions of the complexity of A.

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References

  1. Bednorz, W.: A note on the Men’shov-Rademacher inequality. Bull. Pol. Acad. Sci. 54, 89–93 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bednorz, W.: A theorem on majorizing measures. Ann. Probab. 34, 1771–1781 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bednorz, W.: On Talagrand’s admissible net approach to majorizing measure and boundedness of stochastic processes. Bull. Pol. Acad. Sci. 56, 83–91 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  4. Choobanyan, S., Levental, S., Salehi, H.: On the constant in Men’shov-Rademacher inequality. J. Inequal. Appl. 2006, 1–7 (2006)

    Article  Google Scholar 

  5. Fristedt, B., Gray, L.: A Modern Approach to Probability Theory. Birkhäuser, Boston (1977)

    Google Scholar 

  6. Gaposhkin, V.F.: On convergence of orthogonal series. Dokl. Akad. Nauk SSSR 159, 243–246 (1964) (in Russian)

    MathSciNet  Google Scholar 

  7. Kashin, B.S., Saakyan, A.A.: Orthogonal Series. Translations of Mathematical Monographs, vol. 75. Am. Math. Soc., Providence (1989)

    MATH  Google Scholar 

  8. Olejnik, J.: On a construction of majorizing measures on subsets of \(\Bbb{R}^{n}\) with special metrics. Studia Math., to appear

  9. Paley, R., Zygmund, A.: A note on analytic functions in the unit circle. Proc. Camb. Philos. Soc. 28, 266–272 (1932)

    Article  Google Scholar 

  10. Paszkiewicz, A.: A new proof of the Rademacher-Menshov theorem. Acta Sci. Math. (Szeged) 71, 631–642 (2005)

    MATH  MathSciNet  Google Scholar 

  11. Paszkiewicz, A.: On complete characterization of coefficients of a.e. convergent orthogonal series. Preprint 2008/03, Faculty of Mathematics and Computer Science, University of Łódź

  12. Pisier, G.: Conditions d’entropie assurant la continuité de certains processus. Séminaire d’Analyse Functionnelle 1979–1980. Exposés XIII–XIV. Ecole Polytechnique, Palaiseau

  13. Solodov, A.P.: Concerning an example of Paskiewich. Math. Notes 78, 258–263 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. Talagrand, M.: Regularity of Gaussian processes. Acta Math. 159, 99–149 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  15. Talagrand, M.: Sample boundedness of stochastic processes under increment conditions. Ann. Probab. 18, 1–49 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  16. Talagrand, M.: Upper and Lower Bounds of Stochastic Processes. Springer, Berlin (2005)

    MATH  Google Scholar 

  17. Talagrand, M.: Convergence of orthogonal series using stochastic processes. Preprint

  18. Tandori, K.: Über die Konvergenz der Orthogonalreihen. Acta Sci. Math. (Szeged) 24, 139–151 (1963)

    MATH  MathSciNet  Google Scholar 

  19. Weber, M.: Some theorems related to almost sure convergence of orthogonal series. Indag. Math., N.S. 11, 293–311 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  20. Weber, M.: Uniform bounds under increment conditions. Trans. Am. Math. Soc. 358, 911–936 (2005)

    Article  Google Scholar 

  21. Bednorz, W.: Majorizing measures on metric spaces. C. R., Math., to appear

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  1. Faculty of Mathematics and Computer Science, University of Łódź, Banacha 22, 90-238, Łódź, Poland

    Adam Paszkiewicz

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  1. Adam Paszkiewicz
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Correspondence to Adam Paszkiewicz.

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Paszkiewicz, A. A complete characterization of coefficients of a.e. convergent orthogonal series and majorizing measures. Invent. math. 180, 55–110 (2010). https://doi.org/10.1007/s00222-009-0226-2

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  • DOI: https://doi.org/10.1007/s00222-009-0226-2

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