Rademacher–Menchov theorem

In mathematical analysis, the Rademacher–Menchov theorem, introduced by Rademacher (1922) and Menchoff (1923), gives a sufficient condition for a series of orthogonal functions on an interval to converge almost everywhere.

Statement

If the coefficients c_ν of a series of bounded orthogonal functions on an interval satisfy

$\sum |c_{\nu }{|}^2\log (\nu {)}^2<\mathrm{\infty }$

then the series converges almost everywhere.

References

• Menchoff, D. (1923), "Sur les séries de fonctions orthogonales. (Première Partie. La convergence.)." (in French), Fundamenta Mathematicae 4: 82–105, doi:10.4064/fm-4-1-82-105, ISSN 0016-2736, http://matwbn.icm.edu.pl/tresc.php?wyd=1&tom=4 
• Rademacher, Hans (1922), "Einige Sätze über Reihen von allgemeinen Orthogonalfunktionen", Mathematische Annalen (Springer Berlin / Heidelberg) 87: 112–138, doi:10.1007/BF01458040, ISSN 0025-5831 
• Zygmund, A. (2002), Trigonometric Series. Vol. I, II, Cambridge Mathematical Library (3rd ed.), Cambridge University Press, ISBN 978-0-521-89053-3