# Compatibility of summation methods

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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)