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)