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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 \ref{1} converges absolutely and uniformly on the entire real axis. However, the property of the absolute convergence set of the series \ref{1} being of positive measure, which according to A. Denjoy and N.N. Luzin is sufficient for the series \ref{2} to converge, is not necessary. There exist, for example, perfect sets of measure zero, the absolute convergence on which of the series \ref{1} entails the convergence of the series \ref{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 [[#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, see ''e.g.'' {{Cite|3}} and {{Cite|a1}}, Chapt. 6.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Denjoy,   "Sur l'absolue convergence des séries trigonométriques"  ''C.R. Acad. Sci.'' , '''155'''  (1912)  pp. 135–136</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.N. Luzin,   ''Mat. Sb.'' , '''28'''  (1912)  pp. 461–472</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></table>
+
* {{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}}
====Comments====
 
For generalizations see, e.g., [[#References|[a1]]], Chapt. 6.
 
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Zygmund,   "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)</TD></TR></table>
 

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=32928
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