Skip to main content
SpringerLink
Log in

How Summable are Rademacher Series?

  • Conference paper
Vector Measures, Integration and Related Topics

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 201))

Abstract

Khintchin inequalities show that a.e. convergent Rademacher series belong to all spaces L p([0, 1]), for finite p. In 1975 Rodin and Semenov considered the extension of this result to the setting of rearrangement invariant spaces. The space L N of functions having square exponential integrability plays a prominent role in this problem.

Another way of gauging the summability of Rademacher series is considering the multiplicator space of the Rademacher series in a rearrangement invariant space X, that is,

$$ \Lambda (\mathcal{R}, X) : = \left\{ {f:[0,1] \to \mathbb{R}: f \cdot \sum {\alpha _n r_n } \in X, for all \sum {\alpha _n r_n } \in X } \right\}. $$

. The properties of the space Λ(R, X) are determined by its relation with some classical function spaces (as L N and L ∞([0, 1])) and by the behavior of the logarithm in the function space X.

In this paper we present an overview of the topic and the results recently obtained (together with Sergey V. Astashkin, from the University of Samara, Russia, and Vladimir A. Rodin, from the State University of Voronezh, Russia.)

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. R. Anantharaman and J. Diestel, Sequences in the range of a vector measure, Department of Mathematical Sciences, Kent State University (1989).

    Google Scholar 

  2. R. Anantharaman and J. Diestel, Sequences in the range of a vector measure, Comment. Math. Prace Mat. 30 (1991), 221–235.

    MATH  MathSciNet  Google Scholar 

  3. S.V. Astashkin, About interpolation of subspaces of rearrangement invariant spaces generated by Rademacher system, Int. J. Math. Math. Sci. 25 (2001), 451–465.

    Article  MATH  MathSciNet  Google Scholar 

  4. S.V. Astashkin, On multiplicator space generated by the Rademacher system, Math. Notes 75 (2004), 158–165.

    Article  MATH  MathSciNet  Google Scholar 

  5. S.V. Astashkin and G.P. Curbera, Symmetric kernel of Rademacher multiplicator spaces, J. Funct. Anal. 226 (2005), 173–192.

    Article  MATH  MathSciNet  Google Scholar 

  6. S.V. Astashkin and G.P. Curbera, Rademacher multiplicator spaces equal to L ∞, Proc. Amer. Math. Soc. 136 (2008) 3493–3501.

    Article  MATH  MathSciNet  Google Scholar 

  7. S.V. Astashkin and G.P. Curbera, Rearrangement invariance of Rademacher multiplicator spaces, J. Funct. Anal. 256 (2009), 4071–4094.

    Article  MATH  MathSciNet  Google Scholar 

  8. R.G. Bartle, N. Dunford and J. Schwartz, Weak compactness and vector measures, Canad. J. Math. 7 (1955), 289–305.

    MATH  MathSciNet  Google Scholar 

  9. C. Bennett and K. Rudnick, On Lorentz-Zygmund spaces, Dissert. Math. 175 (1980) 1–67.

    MathSciNet  Google Scholar 

  10. C. Bennett and R. Sharpley, Interpolation of Operators (Academic Press, Boston, 1988).

    MATH  Google Scholar 

  11. G.P. Curbera, Operators into L 1 of a vector measure and applications to Banach lattices, Math. Ann. 293 (1992), 317–330.

    Article  MATH  MathSciNet  Google Scholar 

  12. G.P. Curbera, A note on function spaces generated by Rademacher series, Proc. Edinburgh. Math. Soc. 40 (1997), 119–126.

    Article  MATH  MathSciNet  Google Scholar 

  13. G.P. Curbera and V.A. Rodin, Multiplication operators on the space of Rademacher series in rearrangement invariant spaces, Math. Proc. Cambridge Phil. Soc. 134 (2003), 153–162.

    Article  MATH  MathSciNet  Google Scholar 

  14. A. Khintchin, Über dyadische Brüche, Math. Z. 18 (1923), 109–116.

    Article  MathSciNet  Google Scholar 

  15. A. Khintchin and A.N. Kolmogorov, Über Konvergenz von Reihen, deren Glieder durch den Zufall bestimmt werden, Math Sbornik 32 (1925), 668–677.

    Google Scholar 

  16. M.A. Krasnosel’skii and Ya.B. Rutickii, Convex Functions and Orlicz Spaces, Noordhoff, Gröningen, 1961.

    Google Scholar 

  17. S.G. Krein, Ju.I. Petunin and E.M. Semenov, Interpolation of Linear Operators, Amer. Math. Soc., Providence R.I., 1982.

    Google Scholar 

  18. R. Latala and K. Oleszkiewicz, On the best constant in the Khinchin-Kahane in-equality, Studia Math. 109 (1994), 101–104.

    MATH  MathSciNet  Google Scholar 

  19. J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces vol. II, Springer-Verlag, Berlin, 1979.

    Google Scholar 

  20. D.R. Lewis, Integration with respect to vector measures, Pacific J. Math. 33 (1970), 157–165.

    MATH  MathSciNet  Google Scholar 

  21. G.G. Lorentz, Relations between function spaces, Proc. Amer. Math. Soc. 12 (1961), 127–132.

    Article  MATH  MathSciNet  Google Scholar 

  22. G. Mockenhaupt, W.J. Ricker, Optimal extension of the Hausdorff-Young inequality, J. Reine Angew. Math. 620 (2008), 195–211.

    MATH  MathSciNet  Google Scholar 

  23. S.J. Montgomery-Smith, The distribution of Rademacher sums, Proc. Amer. Math. Soc. 109 (1990) 517–522.

    Article  MATH  MathSciNet  Google Scholar 

  24. S. Okada, W.J. Ricker and E.A. Sánchez Pérez. Optimal Domain and Integral Extension of Operators acting in Function Spaces, Operator Theory Advances Applications, Birkhäuser Verlag, Basel-Berlin-Boston, 2008.

    MATH  Google Scholar 

  25. R.E.A.C. Paley and A. Zygmund, On some series of functions (I), Proc. Camb. Phil. Soc. 26 (1930), 337–357.

    Article  Google Scholar 

  26. H. Rademacher, Einige Sätze über Reihen von allgemeinen Orthogonalfunktionen, Math. Ann. 87 (1922), 111–138.

    Article  MathSciNet  Google Scholar 

  27. V.A. Rodin and E.M. Semenov, Rademacher series in symmetric spaces, Anal. Math. 1 (1975), 207–222.

    Article  MathSciNet  Google Scholar 

  28. V.A. Rodin and E.M. Semenov, The complementability of a subspace that is generated by the Rademacher system in a symmetric space, Functional Anal. Appl. 13 (1979), 150–151.

    Article  MATH  MathSciNet  Google Scholar 

  29. A. Zygmund, Trigonometric series, Cambridge University Press, Cambridge, 1977.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Facultad de Matemáticas, Universidad de Sevilla, Aptdo. 1160, E-41080, Sevilla, Spain

    Guillermo P. Curbera

Authors
  1. Guillermo P. Curbera
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Facultad de Matemáticas, Universidad de Sevilla, 41080, Sevilla, Spain

    Guillermo P. Curbera

  2. Fachbereich 6: Mathematik, Universität Siegen, 57068, Siegen, Germany

    Gerd Mockenhaupt

  3. Mathematisch-Geographische Fakultät, Katholische Universität Eichstätt-Ingolstadt, 85072, Eichstätt, Germany

    Werner J. Ricker

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Birkhäuser Verlag Basel/Switzerland

About this paper

Cite this paper

Curbera, G.P. (2009). How Summable are Rademacher Series?. In: Curbera, G.P., Mockenhaupt, G., Ricker, W.J. (eds) Vector Measures, Integration and Related Topics. Operator Theory: Advances and Applications, vol 201. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0211-2_13

Download citation

Publish with us

Policies and ethics

Search

Navigation