Difference between revisions of "Denjoy-Luzin theorem"
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If the trigonometric series | If the trigonometric series | ||
| − | $$\frac{a_0}{2}+\sum_{n=1}^\infty a_n\cos nx+b_n\sin nx\tag{1}$$ | + | $$\frac{a_0}{2}+\sum_{n=1}^\infty a_n\cos nx+b_n\sin nx\label{1}\tag{1}$$ |
converges absolutely on a set of positive Lebesgue measure, then the series made up of the absolute values of its coefficients, | converges absolutely on a set of positive Lebesgue measure, then the series made up of the absolute values of its coefficients, | ||
| − | $$\frac{|a_0|}{2}+\sum_{n=1}^\infty|a_n|+|b_n|,\tag{2}$$ | + | $$\frac{|a_0|}{2}+\sum_{n=1}^\infty|a_n|+|b_n|,\label{2}\tag{2}$$ |
| − | converges and, consequently, the initial series \ | + | 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 [[#References|[1]]] and by Luzin [[#References|[2]]]; various generalizations of it also exist [[#References|[3]]]. | The theorem was independently established by Denjoy [[#References|[1]]] and by Luzin [[#References|[2]]]; various generalizations of it also exist [[#References|[3]]]. | ||
Revision as of 17:05, 14 February 2020
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 [3].
References
| [1] | A. Denjoy, "Sur l'absolue convergence des séries trigonométriques" C.R. Acad. Sci. , 155 (1912) pp. 135–136 |
| [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) |
Comments
For generalizations see, e.g., [a1], Chapt. 6.
References
| [a1] | A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988) |
Denjoy-Luzin theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Denjoy-Luzin_theorem&oldid=32928