Voronoi summation method
A matrix summation method of sequences. It is defined by a numerical sequence and denoted by the symbol
. A sequence
is summable by the method
to a number
if
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In particular, if ,
,
, the summability of a sequence by the
-method to a number
means that the sequence converges to
. For
,
, one obtains the Cesàro summation method (cf. Cesàro summation methods). For
,
,
, the
-method is regular (cf. Regular summation methods) if and only if
. Any two regular methods
and
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 or
.
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=32749