# 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 | + | {{TEX|done}} |

+ | 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 | + | 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 | + | 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==== | ====References==== |

## 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