Difference between revisions of "Denjoy-Luzin theorem"
From Encyclopedia of Mathematics
(Importing text file) |
m (refs) |
||
| (6 intermediate revisions by 3 users not shown) | |||
| Line 1: | Line 1: | ||
| + | {{TEX|done}} | ||
''on absolutely convergent trigonometric series'' | ''on absolutely convergent trigonometric series'' | ||
If the 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, | + | 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 | + | 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]]] | + | The theorem was independently established by Denjoy [[#References|[1]]] and by Luzin [[#References|[2]]]. Various generalizations of it also exist, see ''e.g.'' {{Cite|3}} and {{Cite|a1}}, Chapt. 6. |
====References==== | ====References==== | ||
| − | + | * {{Ref|1}} A. Denjoy, "Sur l'absolue convergence des séries trigonométriques" ''C.R. Acad. Sci.'' , '''155''' (1912) pp. 135–136 {{ZBL|43.0319.01}} | |
| − | + | * {{Ref|2}} N.N. Luzin, ''Mat. Sb.'' , '''28''' (1912) pp. 461–472 | |
| − | + | * {{Ref|3}} N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian) | |
| − | + | * {{Ref|a1}} A. Zygmund, "Trigonometric series" , '''1–2''' , Cambridge Univ. Press (1988) {{ZBL|0628.42001}} | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
Latest revision as of 12:11, 19 March 2023
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
How to Cite This Entry:
Denjoy-Luzin theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Denjoy-Luzin_theorem&oldid=17015
Denjoy-Luzin theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Denjoy-Luzin_theorem&oldid=17015
This article was adapted from an original article by L.D. KudryavtsevE.M. Nikishin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article