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Lebesgue summation method

From Encyclopedia of Mathematics
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A method for summing trigonometric series. The series

(*)

is summable at a point by the Lebesgue summation method to the sum if in some neighbourhood of this point the integrated series

converges and its sum has symmetric derivative at equal to :

The last condition can also be represented in the form

The Lebesgue summation method is not regular, in the sense that it is not possible to sum every convergent trigonometric series (*) (see Regular summation methods), but if (*) is the Fourier series of a summable function , then it is summable almost-everywhere to by the Lebesgue summation method. The method was proposed by H. Lebesgue [1].

References

[1] H. Lebesgue, "Leçons sur les séries trigonométriques" , Gauthier-Villars (1906)
[2] N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian)
How to Cite This Entry:
Lebesgue summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lebesgue_summation_method&oldid=33421
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article