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 $\{p_n\}$ 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 | 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 | ||
| − | $$\frac{s_0p_n+s_1p_{n-1}+\ | + | $$\frac{s_0p_n+s_1p_{n-1}+\dotsb+s_np_0}{p_0+\dotsb+p_n}\to S$$ |
| − | In particular, if $p_0=1$, $p_k=0$, $k\geq1$, the summability of a sequence by the $(W,p_n)$-method to a number $S$ means that the sequence converges to $S$. For $p_k=1$, $k\geq0$, one obtains the Cesàro summation method (cf. [[Cesàro summation methods|Cesàro summation methods]]). For $p_0>0$, $p_k\geq1$, $k\geq1$, the $(W,p_n)$-method is regular (cf. [[Regular summation methods|Regular summation methods]]) if and only if $p_n/(p_0+\ | + | In particular, if $p_0=1$, $p_k=0$, $k\geq1$, the summability of a sequence by the $(W,p_n)$-method to a number $S$ means that the sequence converges to $S$. For $p_k=1$, $k\geq0$, one obtains the Cesàro summation method (cf. [[Cesàro summation methods|Cesàro summation methods]]). For $p_0>0$, $p_k\geq1$, $k\geq1$, the $(W,p_n)$-method is regular (cf. [[Regular summation methods|Regular summation methods]]) if and only if $p_n/(p_0+\dotsb+p_n)\to0$. Any two regular methods $(W,p_n')$ and $(W,p_n'')$ are compatible (cf. [[Compatibility of summation methods|Compatibility of summation methods]]). |
The Voronoi summation method was first introduced by G.F. Voronoi [[#References|[1]]] and was rediscovered by N.E. Nörlund in 1919. The method is therefore sometimes referred to in western literature as the Nörlund method and the symbol given to it is $(N,p_n)$ or $N(p_n)$. | The Voronoi summation method was first introduced by G.F. Voronoi [[#References|[1]]] and was rediscovered by N.E. Nörlund in 1919. The method is therefore sometimes referred to in western literature as the Nörlund method and the symbol given to it is $(N,p_n)$ or $N(p_n)$. | ||
Latest revision as of 13:40, 14 February 2020
A matrix summation method of sequences. It is defined by a numerical sequence $\{p_n\}$ and denoted by the symbol $(W,p_n)$. A sequence $\{s_n\}$ is summable by the method $(W,p_n)$ to a number $S$ if
$$\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) |
Voronoi summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Voronoi_summation_method&oldid=44631