A convergence result for unconditional series in Lp(μ)

Juan M. Medina, Bruno Cernuschi-Frías


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

Palabras clave

Unconditional basic sequence ; Almost sure convergence ; Random series.

Texto completo:



Alexits G. , Convergence Problems of Orthogonal Series, Pergamon Press, (1961).

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

Gerre, S., Classical Sequences in Banach Spaces, Marcel Dekker, (1992).

Houdré C. On the almost sure convergnece of series of satationary and related nonstationary variables, Ann. of Prob. Vol. 23 (3), pp. 1204- 1218, (1985).

Kahane J. P. Some Random Series of Functions, Cambridge, (1993).

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

Loéve M., Probability Theory, Vol. I, Springer Verlag, (1977).

Medina J. M. , Cernuschi -Frías B. Random series in Lp(X, Σ, µ) 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).

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

Móricz F., Tándori K. An Improved Menshov-Rademacher Theorem, Proc. A. M. S. Vol. 124 (3), pp. 877-885, (1996).

Ørno P. A note on Unconditionally converging series in ¨ Lp, Proc. A. M. S. Vol. 59 (2), 252-254, (1976). Lecture Notes in Mathematics No. 672, Springer-Verlag, (1978).

Wojtaszczyk P. Banach Spaces for Analysts, Cambridge, (1996).

Yang L., Unconditional Basic Sequence in Lp(µ) and its l p stability, Proc. A. M. S. Vol. 127(2), pp. 455-464, (1999).

Zygmund A., Trigonometric Series, Vol II. Cambridge, (1958).

DOI: http://dx.doi.org/10.4067/S0716-09172013000400001

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