Du Bois-Reymond theorem

on the uniqueness of the expansion of a function into a series

If the sum of an everywhere-convergent trigonometric series is Riemann integrable, this series is its Fourier series; demonstrated by P. du Bois-Reymond [1]. G. Cantor [2] discussed at a somewhat earlier date the important special case of trigonometric series converging to zero.

The du Bois-Reymond theorem was generalized in various directions. H. Lebesgue demonstrated a similar theorem for Lebesgue integrals with the condition of boundedness of the sum, while Ch.J. de la Vallée-Poussin [3] demonstrated its validity without this restriction. Analogues of the theorem for Denjoy integrals also exist [5].

In another type of generalization, the condition of everywhere convergence was weakened. W.H. Young (cf. [3]) showed that a countable set may be neglected; D.E. Men'shov (cf. [3]) showed that it is not permissible to discard an arbitrary set of measure zero (cf. Men'shov example of a zero-series). For other studies on this subject see [3], [4]. If the requirement of convergence is replaced by the requirement of summability, yet another kind of generalization is obtained; M. Riesz (cf. [4]) was the first to study this matter.

References

[1]	P. du Bois-Raymond, Abh. Akad. Wiss. Berlin Math.-Phys. Kl. , 12 : 1 (1876) pp. 117–166
[2]	G. Cantor, "Ueber die Ausdehnung eines Satzes aus der Theorie der trigonometrische Reihen" Math. Ann. , 5 (1872) pp. 123–132
[3]	N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian)
[4]	A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988)
[5]	I.A. Vinogradova, V.A. Skvortsov, "Generalized Fourier integrals and series" J. Soviet Math. , 1 : 6 (1973) pp. 677–703 Itogi Nauk. Mat. Anal. 1970 (1971) pp. 65–107