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 . ${\displaystyle X\subseteq \mathbb {R} }$ ${\displaystyle \lambda (X)}$

Keep going

${\displaystyle s(x)={\frac {a_{0}}{2}}+\sum _{k=1}^{\infty }\left(a_{k}\cos(kx)+b_{k}\sin(kx)\right)\;(x\in X)}$

a trigonometric series based on sequences of coefficients consisting of real numbers and . ${\displaystyle X}$ ${\displaystyle \left(a_{k}\right)_{k=0,1,2,3,\ldots }}$ ${\displaystyle \left(b_{k}\right)_{k=1,2,3,\ldots }}$

Then:

It is necessary and sufficient for the absolute convergence of the series that the two associated series of coefficients ${\displaystyle s(x)}$

${\displaystyle A=\sum _{k=1}^{\infty }{a_{k}}}$

and

${\displaystyle B=\sum _{k=1}^{\infty }{b_{k}}}$

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: ${\displaystyle s(x)}$ ${\displaystyle A}$ ${\displaystyle B}$

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 . ${\displaystyle X\subseteq \mathbb {R} }$ ${\displaystyle f\colon X\to \mathbb {R} }$ ${\displaystyle {\lambda }^{*}<\lambda (X)}$ ${\displaystyle X^{*}\subseteq X}$ ${\displaystyle \lambda (X^{*})>{\lambda }^{*}}$ ${\displaystyle f|_{X^{*}}}$

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. ${\displaystyle s(x)}$ ${\displaystyle I\subseteq X}$ ${\displaystyle s(x)}$ ${\displaystyle X}$

(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. ${\displaystyle s(x)}$ ${\displaystyle X\subseteq \mathbb {R} }$ ${\displaystyle s(x)}$

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. ${\displaystyle x\in X}$ ${\displaystyle s(x)}$ ${\displaystyle \left(a_{k}\right)_{k=0,1,2,3,\ldots }}$ ${\displaystyle \left(b_{k}\right)_{k=1,2,3,\ldots }}$ ${\displaystyle s(x)}$ ${\displaystyle X}$

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 ).

Individual evidence

1. ↑ Francesco Giacomo Tricomi: Lectures on orthogonal series. 1970, p. 77 ff
2. ^ Antoni Zygmund: Trigonometric Series. Vol. I. 1977, p. 232 ff
3. ↑ ^{a b} Tricomi, op. Cit. , P. 77
4. ↑ Tricomi, op. Cit. , P. 79
5. ↑ Tricomi, op. Cit. , P. 80
6. ↑ Tricomi, op. Cit. , Pp. 105-106
7. ↑ Zygmund, op. Cit. , P. 316