# Summability multipliers

Numerical factors $\lambda _ {n}$( for the terms of a series) that transform a series

$$\tag{1 } \sum _ { n= } 1 ^ \infty u _ {n}$$

which is summable by a summation method $A$( cf. Summation methods) into a series

$$\tag{2 } \sum _ { n= } 1 ^ \infty \lambda _ {n} u _ {n}$$

which is summable by a method $B$. In this case, the summability multipliers $\lambda _ {n}$ are called summability multipliers of type $( A, B)$. For example, the numbers $\lambda _ {n} = 1/( n+ 1) ^ {s}$ are summability multipliers of type $(( C, k), ( C, k- s))$( see Cesàro summation methods) when $0 < s < k+ 1$( see [1]).

The fundamental problem in the theory of summability multipliers is to find conditions under which numbers $\lambda _ {n}$ will be summability multipliers of one type or another. This question is formulated more exactly in the following way: If $X$ and $Y$ are two classes of series, then what conditions have to be imposed on the numbers $\lambda _ {n}$ so that for every series (1) from $X$, the series (2) belongs to $Y$? The appearance of the theory of summability multipliers goes back to the Dedekind–Hadamard theorem: The series (2) converges for any convergent series (1) if and only if

$$\sum _ { n= } 0 ^ \infty | \Delta \lambda _ {n} | < \infty ,$$

where $\Delta \lambda _ {n} = \lambda _ {n} - \lambda _ {n+} 1$. There is a generalization of this theorem with summability by the Cesàro method.

#### References

 [1] G.H. Hardy, "Divergent series" , Clarendon Press (1949) [2] G.F. Kangro, "On summability factors" Uchen. Zapiski Tartusk. Univ. , 37 (1955) pp. 191–232 (In Russian) [3] G.F. Kangro, "Theory of summability of sequences and series" J. Soviet Math. , 5 : 1 (1970) pp. 1–45 Itogi Nauk. i Tekhn. Mat. Anal. , 12 (1974) pp. 5–70 [4] S.A. Baron, "Introduction to the theory of summability of series" , Tartu (1966) (In Russian) [5] C.N. Moore, "Summable series and convergence factors" , Dover, reprint (1966)
How to Cite This Entry:
Summability multipliers. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Summability_multipliers&oldid=48906
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article