Compatibility of summation methods
A property of summation methods that describes the consistency of the results of applying these methods. Two methods and
are compatible if they cannot sum the same sequence or series to different limits, otherwise they are called incompatible summation methods. More precisely, let
and
be summation methods of sequences say, and let
and
be their summability fields. Then
and
are compatible if
![]() | (*) |
for any , where
and
are the numbers to which
is summed by
and
, respectively. For example, all the Cesàro summation methods
are compatible for
, and so is every regular Voronoi summation method.
If is some set of sequences and
for every
, then one says that
and
are compatible on
.
and
are said to be completely compatible (for real sequences) if (*) also holds in case one includes in their summability fields the sequences summable by these methods to
and
.
References
[1] | G.H. Hardy, "Divergent series" , Clarendon Press (1949) |
[2] | R.G. Cooke, "Infinite matrices and sequence spaces" , Macmillan (1950) |
Comments
The Voronoi summation methods are in the Western literature called Nörlund summation methods, despite the fact that G.F. Voronoi was the first to introduce these methods (1901). N.E. Nörlund later (1920), independently, rediscovered them.
References
[a1] | K. Zeller, W. Beekmann, "Theorie der Limitierungsverfahren" , Springer (1970) |
Compatibility of summation methods. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Compatibility_of_summation_methods&oldid=46414