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Voronoi summation method

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A matrix summation method of sequences. It is defined by a numerical sequence and denoted by the symbol (W,p_n). A sequence \{s_n\} is summable by the method (W,p_n) to a number S if

\frac{s_0p_n+s_1p_{n-1}+\dotsb+s_np_0}{p_0+\dotsb+p_n}\to S

In particular, if p_0=1, p_k=0, k\geq1, the summability of a sequence by the (W,p_n)-method to a number S means that the sequence converges to S. For p_k=1, k\geq0, one obtains the Cesàro summation method (cf. Cesàro summation methods). For p_0>0, p_k\geq1, k\geq1, the (W,p_n)-method is regular (cf. Regular summation methods) if and only if p_n/(p_0+\dotsb+p_n)\to0. Any two regular methods (W,p_n') and (W,p_n'') are compatible (cf. Compatibility of summation methods).

The Voronoi summation method was first introduced by G.F. Voronoi [1] and was rediscovered by N.E. Nörlund in 1919. The method is therefore sometimes referred to in western literature as the Nörlund method and the symbol given to it is (N,p_n) or N(p_n).

References

[1] G.F. [G.F. Voronoi] Woronoi, "Extension of the notion of the limit of the sum of terms of an infinite series" Ann. of Math. (2) , 33 (1932) pp. 422–428 ((With notes by J.D. Tamarkin.))
[2] G.H. Hardy, "Divergent series" , Clarendon Press (1949)


Comments

References

[a1] C.N. Moore, "Summable series and convergence factors" , Dover, reprint (1966)
How to Cite This Entry:
Voronoi summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Voronoi_summation_method&oldid=44631
This article was adapted from an original article by F.I. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article