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Difference between revisions of "Voronoi summation method"

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A [[Matrix summation method|matrix summation method]] of sequences. It is defined by a numerical sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096930/v0969301.png" /> and denoted by the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096930/v0969302.png" />. A sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096930/v0969303.png" /> is summable by the method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096930/v0969305.png" /> to a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096930/v0969306.png" /> if
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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
  
<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/v/v096/v096930/v0969307.png" /></td> </tr></table>
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$$\frac{s_0p_n+s_1p_{n-1}+\dotsb+s_np_0}{p_0+\dotsb+p_n}\to S$$
  
In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096930/v0969308.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096930/v0969309.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096930/v09693010.png" />, the summability of a sequence by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096930/v09693011.png" />-method to a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096930/v09693012.png" /> means that the sequence converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096930/v09693013.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096930/v09693014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096930/v09693015.png" />, one obtains the Cesàro summation method (cf. [[Cesàro summation methods|Cesàro summation methods]]). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096930/v09693016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096930/v09693017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096930/v09693018.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096930/v09693019.png" />-method is regular (cf. [[Regular summation methods|Regular summation methods]]) if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096930/v09693020.png" />. Any two regular methods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096930/v09693021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096930/v09693022.png" /> are compatible (cf. [[Compatibility of summation methods|Compatibility of summation methods]]).
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096930/v09693023.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096930/v09693024.png" />.
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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 $N(p_n)$.
  
 
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Latest revision as of 13:40, 14 February 2020

A 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). 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=18737
This article was adapted from an original article by F.I. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article