Denjoy-Luzin theorem

on absolutely convergent trigonometric series

If the trigonometric series

$$\frac{a_0}{2}+\sum_{n=1}^\infty a_n\cos nx+b_n\sin nx\label{1}\tag{1}$$

converges absolutely on a set of positive Lebesgue measure, then the series made up of the absolute values of its coefficients,

$$\frac{|a_0|}{2}+\sum_{n=1}^\infty|a_n|+|b_n|,\label{2}\tag{2}$$

converges and, consequently, the initial series \eqref{1} converges absolutely and uniformly on the entire real axis. However, the property of the absolute convergence set of the series \eqref{1} being of positive measure, which according to A. Denjoy and N.N. Luzin is sufficient for the series \eqref{2} to converge, is not necessary. There exist, for example, perfect sets of measure zero, the absolute convergence on which of the series \eqref{1} entails the convergence of the series \eqref{2}.

The theorem was independently established by Denjoy [1] and by Luzin [2]. Various generalizations of it also exist, see e.g. [3] and [a1], Chapt. 6.

References

• [1] A. Denjoy, "Sur l'absolue convergence des séries trigonométriques" C.R. Acad. Sci. , 155 (1912) pp. 135–136 Zbl 43.0319.01
• [2] N.N. Luzin, Mat. Sb. , 28 (1912) pp. 461–472
• [3] N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian)
• [a1] A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988) Zbl 0628.42001