Matrix summation method

One of the methods for summing series and sequences using an infinite matrix. Employing an infinite matrix , a given sequence is transformed into the sequence :

If the series on the right-hand side converges for all and if the sequence has a limit for :

then the sequence is said to be summable by the method determined by the matrix , or simply summable by the matrix , and the number is referred to as its limit in the sense of this summation method. If is regarded as the sequence of partial sums of a series

(1)

then this series is said to be summable to the sum by the matrix .

A matrix summation method for series can be also defined directly by transforming the series (1) into a sequence :

(2)

where is a given matrix. In this case the series (1) is said to be summable to the sum if, for all the series on the right-hand side in (2) converges and

Less often used are matrix summation methods defined by a transformation of a series (1) into a series

(3)

where

or by a transformation of a sequence into a series

(4)

where

which use matrices and , respectively. In these cases the series (1) with the partial sums is summable to the sum if the series (3) converges to or, respectively, if the series (4) converges to .

The matrix of a summation method all entries of which are non-negative is called a positive matrix. Among the matrix summation methods one finds, for example, the Voronoi summation method, the Cesàro summation methods, the Euler summation method, the Riesz summation method , the Hausdorff summation method, and others (see also Summation methods).

References

[1]	G.H. Hardy, "Divergent series" , Clarendon Press (1949)
[2]	R.G. Cooke, "Infinite matrices and sequence spaces" , Macmillan (1950)
[3]	G.P. Kangro, "Theory of summability of sequences and series" J. Soviet Math. , 5 : 1 (1976) 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)