Lebesgue summation method

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 $s$:

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

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 $\eqref{*}$ (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].

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)