A convergence result for unconditional series in L^p(μ)

 

Juan M. Medina, Bruno Cernuschi-Frías
Universidad de Buenos Aires, Argentina

---

ABSTRACT

We give sufficient conditions for the convergence almost everywhere of the expansion with respect to an unconditional basis for functions in L^p p > 2. This result extends the classical theorem of Menchoff and Rademacher for orthogonal series in L².

Subjclass 2000 MSC Primary 42C15 ; Secondary: 46B15, 46B20.

Keywords : Unconditional basic sequence, almost sure convergence, random series.

---

 

REFERENCES

1 Alexits G. , Convergence Problems of Orthogonal Series, Pergamon Press, (1961).         [ Links ]

2 Bennett, G. Unconditional Convergence and Almost Everywhere Convergence Z. Wahrs. verw. Gebeite Vol. 34, pp. 135-155, (1976).         [ Links ]

3 Gerre, S., Classical Sequences in Banach Spaces, Marcel Dekker, (1992).         [ Links ]

4 Houdre C. On the almost sure convergnece of series of satationary and related nonstationary variables, Ann. of Prob. Vol. 23 (3), pp. 12041218, (1985).         [ Links ]

5 Kahane J. P. Some Random Series of Functions, Cambridge, (1993).         [ Links ]

6 Lindenstrauss J. Tzafriri L. Classical Banach Spaces, Vol. I y II, Springer Verlag 2ed., (1996).         [ Links ]

7 Loeve M., Probability Theory, Vol. I, Springer Verlag, (1977).         [ Links ]

8 MedinaJ.M., Cernuschi-Frias B. Random series in L^p(X, E,/)using Unconditional Basic Sequences and l^p stable sequences: A result on almost sure almost everywhere convergence, Proc. A. M. S. Vol.135(11), pp. 3561-3569, (2007).         [ Links ]

9 Menchoff D. Sur les series de fonctions orthogonales I., Fund. Math. 4,1923, pages 82-105. Vol. 40 (2), September, pp. 1490-1503, (1994).         [ Links ]

10 Moiricz F., Tiandori K. An Improved Menshov-Rademacher Theorem, Proc. A. M. S. Vol. 124 (3), pp. 877-885, (1996).         [ Links ]

11 0rno P. A note on Unconditionally converging series in L^p, Proc. A.M. S. Vol. 59 (2), 252-254, (1976). Lecture Notes in Mathematics N°.672, Springer-Verlag, (1978).         [ Links ]

12 Wojtaszczyk P. Banach Spaces for Analysts, Cambridge, (1996).         [ Links ]

13 Yang L., Unconditional Basic Sequence in L^p (/)and its l^p stability, Proc. A. M. S. Vol. 127(2), pp. 455-464, (1999).         [ Links ]

14 Zygmund A., Trigonometric Series, Vol II. Cambridge, (1958).A convergence result for unconditional series in L^p(/)319        [ Links ]

 

Juan M. Medina
Departamento de Matematica,
Instituto Argentino de Matematica,
Universidad de Buenos Aires,
Paseo Colion 850 (1063)
Capital Federal
CONICET
Argentina
e-mail : jmedina@fi.uba.ar

 

Bruno Cernuschi—Frías
Facultad de Ingeniería
Instituto Argentino de Matematica
Universidad de Buenos Aires
Paseo Colion 850 (1063)
Capital Federal
CONICET
Argentina
e-mail : bcf@ieee.org

*This work was partially supported by the Universidad de Buenos Aires, grant No. UBACyT 20020100100503, and the Consejo Nacional de Investigaciones Científicas y Tecnicas, CONICET, Argentina.

Received : May 2012. Accepted : July 2013.