View source for Riemann summation method

A method for summing series of numbers. A series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082030/r0820301.png" /> can be summed by Riemann's method to a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082030/r0820302.png" /> if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082030/r0820303.png" /></td> </tr></table>

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082030/r0820304.png" /></td> </tr></table>

with bounded coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082030/r0820305.png" /> can be summed by Riemann's method at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082030/r0820306.png" /> to a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082030/r0820307.png" /> if the function

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082030/r0820308.png" /></td> </tr></table>

has, at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082030/r0820309.png" />, [[Riemann derivative|Riemann derivative]] equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082030/r08203010.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B. Riemann,   "Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe" , ''Gesammelte Math. Abhandlungen'' , Dover, reprint  (1957)  pp. 227–264</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><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Zygmund,   "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  G.H. Hardy,   "Divergent series" , Clarendon Press  (1949)</TD></TR></table>



====Comments====
Regularity (cf. [[Regular summation methods|Regular summation methods]]) is expressed by Riemann's first theorem; the theorem stated above is called Riemann's second theorem. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082030/r08203011.png" /> is also called the [[Riemann function|Riemann function]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Beekman,   "Theorie der Limitierungsverfahren" , Springer  (1970)</TD></TR></table>