Lebesgue summation method

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A method for summing trigonometric series. The series

$$\frac{a_0}{2}+\sum_{n=1}^\infty a_n\cos nx+b_n\sin nx\label{*}\tag{*}$$

is summable at a point $x_0$ by the Lebesgue summation method to the sum $s$ if in some neighbourhood $(x_0-h,x_0+h)$ of this point the integrated series

$$\frac{a_0x}{2}+\sum_{n=1}^\infty\frac1n(a_n\sin nx-b_n\cos nx)$$

converges and its sum $F(x)$ has symmetric derivative at $x_0$ equal to $s$:


The last condition can also be represented in the form

$$\lim_{h\to0}\left[\frac{a_0}{2}+\sum_{n=1}^\infty(a_n\cos nx_0+b_n\sin nx_0)\frac{\sin nh}{nh}\right]=s.$$

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


[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:
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article