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

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

then the series converges almost everywhere.

References

• Menchoff, D. (1923), "Sur les séries de fonctions orthogonales. (Premiére Partie. La convergence.).", Fundamenta Mathematicae (in French) 4: 82–105, ISSN 0016-2736
• 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) [1935], Trigonometric series. Vol. I, II, Cambridge Mathematical Library (3rd ed.), Cambridge University Press, ISBN 978-0-521-89053-3, MR 1963498