Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

An improved Menshov-Rademacher theorem


Authors: Ferenc Móricz and Károly Tandori
Journal: Proc. Amer. Math. Soc. 124 (1996), 877-885
MSC (1991): Primary 42C05
DOI: https://doi.org/10.1090/S0002-9939-96-03151-6
MathSciNet review: 1301040
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 $[0,1]$ and the moduli of the coefficients form a nonincreasing sequence. We also prove a new version of the Menshov-Rademacher inequality.


References [Enhancements On Off] (What's this?)

  • 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. K. Tandori, Über die Divergenz der Orthogonalreihen, Publ. Math. Debrecen 8 (1961), 291--307. MR 25:2374
  • 7. ------, Orthogonalen Funktionen X, Acta Sci. Math. (Szeged) 23 (1962), 185--221. MR 26:1688
  • 8. ------, Über die Konvergenz der Orthogonalreihen II, Acta Sci. Math. (Szeged) 25 (1964), 219--232. MR 30:1350
  • 9. ------, Bemerkung zur Konvergenz der Orthogonalreihen, Acta Sci. Math. (Szeged) 26 (1965), 249--251. MR 33:3041

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 42C05

Retrieve articles in all journals with MSC (1991): 42C05


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: https://doi.org/10.1090/S0002-9939-96-03151-6
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
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society