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An improved Menshov-Rademacher theorem
Author(s):
Ferenc
Móricz;
Károly
Tandori
Journal:
Proc. Amer. Math. Soc.
124
(1996),
877-885.
MSC (1991):
Primary 42C05
MathSciNet review:
1301040
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Abstract:
We study the a.e. convergence of orthogonal series defined over a general measure space. We give sufficient conditions which contain the Menshov-Rademacher theorem as an endpoint case. These conditions turn out to be necessary in the particular case where the measure space is the unit interval ![$[0,1]$](/proc/1996-124-03/S0002-9939-96-03151-6/gif-abstract/img1.gif) and the moduli of the coefficients form a nonincreasing sequence. We also prove a new version of the Menshov-Rademacher inequality.
References:
- 1.
- G. Alexits, Convergence problems of orthogonal series, Hungarian Acad. Sci., Budapest, 1961. MR 36:1911
- 2.
- D. E. Menchoff, Sur les séries des fonctions orthogonales (Première partie), Fund. Math. 4 (1923), 92--105.
- 3.
- F. Móricz, Moment inequalities and the strong laws of large numbers, Z. Wahrsch. Verw. Gebiete 35 (1976), 299--314. MR 53:11717
- 4.
- F. Móricz and K. Tandori, Almost everywhere convergence of orthogonal series revisited, J. Math. Anal. Appl. 182 (1994), 637--653. MR 95a:42049
- 5.
- H. Rademacher, Einige Sätze über Reihen von allgemeinen Orthogonalfunktionen, Math. Ann. 87 (1922), 112--138.
- 6.
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Additional Information:
Ferenc
Móricz
Affiliation:
Bolyai Institute, University of Szeged, Aradi Vértanúk Tere 1, 6720 Szeged, Hungary
Email:
moricz@math.u-szeged.hu
Károly
Tandori
Affiliation:
Bolyai Institute, University of Szeged, Aradi Vértanúk Tere 1, 6720 Szeged, Hungary
DOI:
10.1090/S0002-9939-96-03151-6
PII:
S 0002-9939(96)03151-6
Keywords:
Orthonormal system,
a.e. convergence,
Menshov-Rademacher inequality and theorem
Received by editor(s):
November 1, 1993
Received by editor(s) in revised form:
September 26, 1994
Additional Notes:
This research was partially supported by the Hungarian National Foundation for Scientific Research under Grant #234
Communicated by:
J. Marshall Ash
Copyright of article:
Copyright
1996,
American Mathematical Society
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