Lusin-Denjoy theorem
The Lusin-Denjoy theorem is one of the classic theorems of the mathematical branch of analysis . It goes back to two papers published side by side in the same specialist journal in 1912, submitted by the mathematicians Nikolai Nikolajewitsch Lusin and Arnaud Denjoy . The theorem deals with and clarifies the important question of the convergence behavior of the real trigonometric series .
Formulation of the Lusin-Denjoy theorem
It can be formulated as follows:
- Let the body of the real numbers be a Lebesgue-measurable point set of positive Lebesgue measure .
- Keep going
- a trigonometric series based on sequences of coefficients consisting of real numbers and .
- Then:
-
It is necessary and sufficient for the absolute convergence of the series that the two associated series of coefficients
-
and
- both absolutely converge.
Note on evidence
In proving the Lusin-Denjoy theorem, as the Italian mathematician Francesco Giacomo Tricomi emphasizes in his lectures on orthogonal series , the real difficulty and the essential proof lies in the proof that - under the above-mentioned conditions! - from the absolute convergence of the given trigonometric series the absolute convergence of the two associated series of coefficients and follows. In this proof step, according to Tricomi, a proposition from real measure theory is significant, which essentially says the following:
- If a Lebesgue-measurable real point set has a positive Lebesgue measure and if a Lebesgue-measurable real function is given on it, there is a Lebesgue-measurable real point set for every given positive real number , which on the one hand has a Lebesgue measure and for the on the other hand the constraint is a limited function .
Immediate conclusions
With the Lusin-Denjoy theorem one immediately gains the following two corollaries :
- (I) If, under the given conditions, the trigonometric series is absolutely convergent even on any interval of positive length , then it is also absolutely absolutely convergent.
- (II) If a trigonometric series converges absolutely on an arbitrary set of points , then it converges absolutely and uniformly on every interval of positive length within it.
Related sentence
Closely related to the Lusin-Denjoy theorem is the Cantor-Lebesgue theorem , which is named after the two mathematicians Georg Cantor and Henri Lebesgue . This theorem takes up the related question to what extent the convergence behavior of a trigonometric series influences the convergence behavior of the associated coefficient sequences. It says namely:
- If the general requirements of the Lusin-Denjoy theorem are fulfilled and if the partial sums of partial sums are all zero sequences , then both coefficient sequences and are also zero sequences. This is particularly the case when the on converge.
literature
- A. Denjoy: Sur l'absolue convergence des séries trigonométriques . In: Comptes Rendus Mathématique. Académie des Sciences. Paris . tape 155 , 1912, pp. 580-582 ( online ).
- N. Lusin: Sur l'absolue convergence des séries trigonométriques . In: Comptes Rendus Mathématique. Académie des Sciences. Paris . tape 155 , 1912, pp. 135-136 .
- Francesco Giacomo Tricomi: Lectures on Orthogonal Series . Translated and edited for printing by Prof. Dr. Friedrich Kasch , Munich (= The basic teachings of the mathematical sciences in individual representations with special consideration of the areas of application . Volume 76 ). 2nd, corrected edition. Springer Verlag , Berlin, Heidelberg, New York 1970 ( MR0261250 ).
- Antoni Zygmund : Trigonometric Series. Volumes I and II . Reprinting of the 1968 Version of the Second Edition. 2nd Edition. Cambridge University Press , Cambridge, London, New York, Melbourne 1977, ISBN 0-521-07477-0 ( MR0617944 ).