Riemann summation method

A method for summing series of numbers. A series $\sum_{n=0}^\infty a_n$ can be summed by Riemann's method to a number $S$ if

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

This method was first introduced and its regularity was first proved by B. Riemann in 1854 (see [1]). The Riemann summation method has been applied in the theory of trigonometric series, where it is usually stated as follows: A trigonometric series

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

with bounded coefficients $a_n,b_n$ can be summed by Riemann's method at a point $x_0$ to a number $S$ if the function

$$F(x)=\frac{a_0x^2}{4}-\sum_{n=1}^\infty\frac{a_n\cos nx+b_n\sin nx}{n^2}$$

has, at $x_0$, Riemann derivative equal to $S$.

References

[1]	B. Riemann, "Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe" , Gesammelte Math. Abhandlungen , Dover, reprint (1957) pp. 227–264
[2]	N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian)
[3]	A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988)
[4]	G.H. Hardy, "Divergent series" , Clarendon Press (1949)

Comments

Regularity (cf. Regular summation methods) is expressed by Riemann's first theorem; the theorem stated above is called Riemann's second theorem. The function $F(x)$ is also called the Riemann function.

References

[a1]	W. Beekman, "Theorie der Limitierungsverfahren" , Springer (1970)