View source for Matrix summation method

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A matrix summation method is one of the methods for summing series and sequences using an infinite matrix. Employing an infinite matrix $[a_{nk}]$, $n,k=1,2,\ldots,$ a given sequence $(s_n)$ is transformed into the sequence $\sigma_n$:
$$
\sigma_n = \sum_{k=1}^\infty a_{nk}s_k.
$$
If the series on the right-hand side converges for all $n=1,2,\ldots,$ and if the sequence $\sigma_n$ has a limit $s$ for $n \rightarrow \infty$:
$$
\lim_{n\rightarrow\infty} \sigma_n = s,
$$
then the sequence $(s_n)$ is said to be summable by the method determined by the matrix $[a_{nk}]$, or simply summable by the matrix $[a_{nk}]$, and the number $s$ is referred to as its limit in the sense of this summation method. If $(s_n)$ is regarded as the sequence of partial sums of a series
\begin{equation}
\label{eq1}
\sum_{k=1}^\infty u_k,
\end{equation}
then this series is said to be summable to the sum $s$ by the matrix 
$[a_{nk}]$.

A matrix summation method for series can be also defined directly by transforming the series \ref{eq1} into a sequence $(\gamma_n)$:
\begin{equation}
\label{eq2}
\gamma_n = \sum_{k=1}^\infty g_{nk}u_k,
\end{equation}
where $[g_{nk}]$ is a given matrix.  In this case the series \ref{eq1} is said to be summable to the sum $s$ if, for all $n=1,2,\ldots,$ the series on the right-hand side in \ref{eq2} converges and
$$
\lim_{n\rightarrow\infty} \gamma_n = s,
$$

Less often used are matrix summation methods defined by a transformation of a series \ref{eq1} into a series
\begin{equation}
\label{eq3}
\sum_{n=1}^\infty \alpha_n,
\end{equation}
where
$$
\alpha_n = \sum_{k=1}^\infty \alpha_{nk}u_k,
$$
or by a transformation of a sequence $(s_n)$ into a series
\begin{equation}
\label{eq4}
\sum_{n=1}^\infty \beta_n,
\end{equation}
where
$$
\beta_n = \sum_{k=1}^\infty \beta_{nk}s_k, \quad n=1,2,\ldots,
$$
which use matrices $[\alpha_{nk}]$ and $[\beta_{nk}]$, respectively. In these cases the series \ref{eq1} with the partial sums $s_n$ is summable to the sum $s$ if the series \ref{eq3} converges to $s$ or, respectively, if the series \ref{eq4} converges to $s$.

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]] $(R,p_n)$, the [[Hausdorff summation method]], and others (see also [[Summation methods]]).

====References==== 

{|
|-
|valign="top"|{{Ref|Ba}}||valign="top"| S.A. Baron, "Introduction to the theory of summability of series", Tartu (1966) (In Russian)
|-
|valign="top"|{{Ref|Co}}||valign="top"| R.G. Cooke, "Infinite matrices and sequence spaces", Macmillan (1950)
|-
|valign="top"|{{Ref|Ha}}||valign="top"| G.H. Hardy, "Divergent series", Clarendon Press (1949)
|-
|valign="top"|{{Ref|Ka}}||valign="top"| 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
|-
|}