A convergence result for unconditional series in Lp(μ)
DOI:
https://doi.org/10.4067/S0716-09172013000400001Keywords:
Unconditional basic sequence, Almost sure convergence, Random series.Abstract
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.References
[1] Alexits G. , Convergence Problems of Orthogonal Series, Pergamon Press, (1961).
[2] Bennett, G. Unconditional Convergence and Almost Everywhere Convergence Z. Wahrs. verw. Gebeite Vol. 34, pp. 135-155, (1976).
[3] Gerre, S., Classical Sequences in Banach Spaces, Marcel Dekker, (1992).
[4] 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).
[5] Kahane J. P. Some Random Series of Functions, Cambridge, (1993).
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[7] Loéve M., Probability Theory, Vol. I, Springer Verlag, (1977).
[8] 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).
[9] Menchoff D. Sur les séries de fonctions orthogonales I., Fund. Math. 4, 1923, pages 82-105. Vol. 40 (2), September, pp. 1490-1503, (1994).
[10] Móricz F., Tándori K. An Improved Menshov-Rademacher Theorem, Proc. A. M. S. Vol. 124 (3), pp. 877-885, (1996).
[11] 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).
[12] Wojtaszczyk P. Banach Spaces for Analysts, Cambridge, (1996).
[13] Yang L., Unconditional Basic Sequence in Lp(µ) and its l p stability, Proc. A. M. S. Vol. 127(2), pp. 455-464, (1999).
[14] Zygmund A., Trigonometric Series, Vol II. Cambridge, (1958).
[2] Bennett, G. Unconditional Convergence and Almost Everywhere Convergence Z. Wahrs. verw. Gebeite Vol. 34, pp. 135-155, (1976).
[3] Gerre, S., Classical Sequences in Banach Spaces, Marcel Dekker, (1992).
[4] 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).
[5] Kahane J. P. Some Random Series of Functions, Cambridge, (1993).
[6] Lindenstrauss J. Tzafriri L. Classical Banach Spaces, Vol. I y II, Springer Verlag 2ed., (1996).
[7] Loéve M., Probability Theory, Vol. I, Springer Verlag, (1977).
[8] 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).
[9] Menchoff D. Sur les séries de fonctions orthogonales I., Fund. Math. 4, 1923, pages 82-105. Vol. 40 (2), September, pp. 1490-1503, (1994).
[10] Móricz F., Tándori K. An Improved Menshov-Rademacher Theorem, Proc. A. M. S. Vol. 124 (3), pp. 877-885, (1996).
[11] 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).
[12] Wojtaszczyk P. Banach Spaces for Analysts, Cambridge, (1996).
[13] Yang L., Unconditional Basic Sequence in Lp(µ) and its l p stability, Proc. A. M. S. Vol. 127(2), pp. 455-464, (1999).
[14] Zygmund A., Trigonometric Series, Vol II. Cambridge, (1958).
How to Cite
[1]
J. M. Medina and B. Cernuschi-Frías, “A convergence result for unconditional series in Lp(μ)”, Proyecciones (Antofagasta, On line), vol. 32, no. 4, pp. 305-319, 1.
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