View source for 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 [[#References|[1]]]. G. Cantor [[#References|[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 [[#References|[3]]] demonstrated its validity without this restriction. Analogues of the theorem for Denjoy integrals also exist [[#References|[5]]].

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

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. du Bois-Raymond,   ''Abh. Akad. Wiss. Berlin Math.-Phys. Kl.'' , '''12''' :  1  (1876)  pp. 117–166</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Cantor,   "Ueber die Ausdehnung eines Satzes aus der Theorie der trigonometrische Reihen"  ''Math. Ann.'' , '''5'''  (1872)  pp. 123–132</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.K. [N.K. Bari] Bary,   "A treatise on trigonometric series" , Pergamon  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A. Zygmund,   "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  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</TD></TR></table>