Compatibility of summation methods

A property of summation methods that describes the consistency of the results of applying these methods. Two methods $ A $ and $ B $ are compatible if they cannot sum the same sequence or series to different limits, otherwise they are called incompatible summation methods. More precisely, let $ A $ and $ B $ be summation methods of sequences say, and let $ A ^ {*} $ and $ B ^ {*} $ be their summability fields. Then $ A $ and $ B $ are compatible if

$$ \tag{* } \overline{A}\; ( x) = \ \overline{B}\; ( x) $$

for any $ x \in A ^ {*} \cap B ^ {*} $, where $ \overline{A}\; ( x) $ and $ \overline{B}\; ( x) $ are the numbers to which $ x $ is summed by $ A $ and $ B $, respectively. For example, all the Cesàro summation methods $ ( C , k ) $ are compatible for $ k > - 1 $, and so is every regular Voronoi summation method.

If $ U $ is some set of sequences and $ \overline{A}\; ( x) = \overline{B}\; ( x) $ for every $ x \in A ^ {*} \cap B ^ {*} \cap U $, then one says that $ A $ and $ B $ are compatible on $ U $. $ A $ and $ B $ 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 $ + \infty $ and $ - \infty $.

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)