View source for Lebesgue summation method

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A method for summing [[Trigonometric series|trigonometric series]]. The series

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

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 $s$:

$$\lim_{h\to0}\frac{F(x_0+h)-F(x_0-h)}{2h}=s.$$

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 \ref{*} (see [[Regular summation methods|Regular summation methods]]), but if \ref{*} is the [[Fourier series|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 [[#References|[1]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Lebesgue,   "Leçons sur les séries trigonométriques" , Gauthier-Villars  (1906)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.K. [N.K. Bari] Bary,   "A treatise on trigonometric series" , Pergamon  (1964)  (Translated from Russian)</TD></TR></table>