Riemann summation method
From Encyclopedia of Mathematics
A method for summing series of numbers. A series can be summed by Riemann's method to a number
if
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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
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with bounded coefficients can be summed by Riemann's method at a point
to a number
if the function
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has, at , Riemann derivative equal to
.
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 is also called the Riemann function.
References
[a1] | W. Beekman, "Theorie der Limitierungsverfahren" , Springer (1970) |
How to Cite This Entry:
Riemann summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemann_summation_method&oldid=33493
Riemann summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemann_summation_method&oldid=33493
This article was adapted from an original article by T.P. Lykashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article