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
:
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If the series on the right-hand side converges for all and if the sequence
has a limit
for
:
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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
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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) |
Matrix summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Matrix_summation_method&oldid=12057