View source for Voronoi summation method

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A [[Matrix summation method|matrix summation method]] of sequences. It is defined by a numerical sequence $\{p_n\}$ 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|Cesàro summation methods]]). For $p_0>0$, $p_k\geq1$, $k\geq1$, the $(W,p_n)$-method is regular (cf. [[Regular summation methods|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|Compatibility of summation methods]]).

The Voronoi summation method was first introduced by G.F. Voronoi [[#References|[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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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.))</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.H. Hardy,   "Divergent series" , Clarendon Press  (1949)</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C.N. Moore,   "Summable series and convergence factors" , Dover, reprint  (1966)</TD></TR></table>