Inclusion of summation methods

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An inclusion of the summability fields (cf. Summability field) corresponding to these methods. Let $A$ and $B$ be two summation methods defined on a set $M$ of series (or sequences); let $A^*$ and $B^*$ be their summability fields and suppose $A^*\subset B^*$ one then says that method $B$ includes method $A$, which is denoted by the symbol $A\subset B$. Methods $A$ and $B$ are said to be equipotent, denoted by $A=B$, if each of them includes the other. Equipotent methods have the same summability field. Method $B$ is said to be stronger than method $A$ if $B$ includes $A$ but is not equipotent with it. If the summability field of the method is identical with the set of all convergent series, the method is said to be equipotent with convergence. An inclusion of summation methods is sometimes considered not on the entire set of their definition, but only on some of its subsets.

For the Cesàro summation methods $(C,k)$ the inclusion $(C,k_1)\subset(C,k_2)$ is valid for $k_2\geq k_1>-1$; the Abel summation method is stronger than all Cesàro methods $(C,k)$ for $k>-1$; the Riesz summation method $(R,n,k)$ is equipotent with the Cesàro summation method $(C,k)$ ($k\geq0$); the Abel summation method is equipotent with convergence on the set of series whose terms $a_n$ satisfy the condition $a_n=O(1/n)$. In these examples the summation methods are compatible as well (cf. Compatibility of summation methods), even though, in general, an inclusion of summation methods does not rest on the assumption of their being compatible. However, if $A$ and $B$ are regular matrix methods (cf. Regular summation methods) and $A\subset B$ on the set of bounded sequences, then $A$ and $B$ are compatible on this set (the Mazur–Orlicz–Brudno theorem). The compatibility requirement is imposed in the very definition of inclusion in certain textbooks.

An inclusion of summation methods defined on a set of series with real terms is said to be complete if the inclusion of their summability fields is preserved after these fields have been completed with series summable to $+\infty$ and $-\infty$. Thus, the Hölder summation method (cf. Hölder summation methods) $(H,k)$ completely includes the Cesàro method $(C,k)$.

An inclusion of summation methods for special types of summability (e.g. absolute summability, strong summability, etc.) is defined in a similar manner.


[1] G.H. Hardy, "Divergent series" , Oxford Univ. Press (1949) Zbl 0032.05801
[2] R.G. Cooke, "Infinite matrices and sequence spaces" , Macmillan (1950)
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[4] S. Mazur, W. Orlicz, "Sur les méthodes linéaires de sommation" C.R. Acad. Sci. Paris Sér. I Math. , 196 (1933) pp. 32–34
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[6] S.A. Baron, "Introduction to the theory of summability of series" , Tartu (1966) (In Russian)
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Inclusion of summation methods. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article