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,
. 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.)
Mathematics Subject Classification (2000)
Keywords
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
R. Anantharaman and J. Diestel, Sequences in the range of a vector measure, Department of Mathematical Sciences, Kent State University (1989).
R. Anantharaman and J. Diestel, Sequences in the range of a vector measure, Comment. Math. Prace Mat. 30 (1991), 221–235.
S.V. Astashkin, About interpolation of subspaces of rearrangement invariant spaces generated by Rademacher system, Int. J. Math. Math. Sci. 25 (2001), 451–465.
S.V. Astashkin, On multiplicator space generated by the Rademacher system, Math. Notes 75 (2004), 158–165.
S.V. Astashkin and G.P. Curbera, Symmetric kernel of Rademacher multiplicator spaces, J. Funct. Anal. 226 (2005), 173–192.
S.V. Astashkin and G.P. Curbera, Rademacher multiplicator spaces equal to L ∞, Proc. Amer. Math. Soc. 136 (2008) 3493–3501.
S.V. Astashkin and G.P. Curbera, Rearrangement invariance of Rademacher multiplicator spaces, J. Funct. Anal. 256 (2009), 4071–4094.
R.G. Bartle, N. Dunford and J. Schwartz, Weak compactness and vector measures, Canad. J. Math. 7 (1955), 289–305.
C. Bennett and K. Rudnick, On Lorentz-Zygmund spaces, Dissert. Math. 175 (1980) 1–67.
C. Bennett and R. Sharpley, Interpolation of Operators (Academic Press, Boston, 1988).
G.P. Curbera, Operators into L 1 of a vector measure and applications to Banach lattices, Math. Ann. 293 (1992), 317–330.
G.P. Curbera, A note on function spaces generated by Rademacher series, Proc. Edinburgh. Math. Soc. 40 (1997), 119–126.
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.
A. Khintchin, Über dyadische Brüche, Math. Z. 18 (1923), 109–116.
A. Khintchin and A.N. Kolmogorov, Über Konvergenz von Reihen, deren Glieder durch den Zufall bestimmt werden, Math Sbornik 32 (1925), 668–677.
M.A. Krasnosel’skii and Ya.B. Rutickii, Convex Functions and Orlicz Spaces, Noordhoff, Gröningen, 1961.
S.G. Krein, Ju.I. Petunin and E.M. Semenov, Interpolation of Linear Operators, Amer. Math. Soc., Providence R.I., 1982.
R. Latala and K. Oleszkiewicz, On the best constant in the Khinchin-Kahane in-equality, Studia Math. 109 (1994), 101–104.
J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces vol. II, Springer-Verlag, Berlin, 1979.
D.R. Lewis, Integration with respect to vector measures, Pacific J. Math. 33 (1970), 157–165.
G.G. Lorentz, Relations between function spaces, Proc. Amer. Math. Soc. 12 (1961), 127–132.
G. Mockenhaupt, W.J. Ricker, Optimal extension of the Hausdorff-Young inequality, J. Reine Angew. Math. 620 (2008), 195–211.
S.J. Montgomery-Smith, The distribution of Rademacher sums, Proc. Amer. Math. Soc. 109 (1990) 517–522.
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.
R.E.A.C. Paley and A. Zygmund, On some series of functions (I), Proc. Camb. Phil. Soc. 26 (1930), 337–357.
H. Rademacher, Einige Sätze über Reihen von allgemeinen Orthogonalfunktionen, Math. Ann. 87 (1922), 111–138.
V.A. Rodin and E.M. Semenov, Rademacher series in symmetric spaces, Anal. Math. 1 (1975), 207–222.
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.
A. Zygmund, Trigonometric series, Cambridge University Press, Cambridge, 1977.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights 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
DOI: https://doi.org/10.1007/978-3-0346-0211-2_13
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0346-0210-5
Online ISBN: 978-3-0346-0211-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)