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Difference between revisions of "Denjoy-Luzin theorem"

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$$\frac{a_0}{2}+\sum_{n=1}^\infty a_n\cos nx+b_n\sin nx\label{1}\tag{1}$$
 
$$\frac{a_0}{2}+\sum_{n=1}^\infty a_n\cos nx+b_n\sin nx\label{1}\tag{1}$$
  
converges absolutely on a set of positive Lebesgue measure, then the series made up of the absolute values of its coefficients,
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converges absolutely on a set of positive [[Lebesgue measure]], then the series made up of the absolute values of its coefficients,
  
 
$$\frac{|a_0|}{2}+\sum_{n=1}^\infty|a_n|+|b_n|,\label{2}\tag{2}$$
 
$$\frac{|a_0|}{2}+\sum_{n=1}^\infty|a_n|+|b_n|,\label{2}\tag{2}$$
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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}.
 
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]]].
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The theorem was independently established by Denjoy [[#References|[1]]] and by Luzin [[#References|[2]]]; various generalizations of it also exist, see ''e.g.'' [[#References|[3]]] and [[#References|[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>
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<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>
====Comments====
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<TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Zygmund, "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)</TD></TR>
For generalizations see, e.g., [[#References|[a1]]], Chapt. 6.
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</table>
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Zygmund,   "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)</TD></TR></table>
 

Revision as of 12:06, 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
[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)
How to Cite This Entry:
Denjoy-Luzin theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Denjoy-Luzin_theorem&oldid=52966
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