Matrix summation method

2020 Mathematics Subject Classification: Primary: 40C05 [MSN][ZBL]

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

[Ba]	S.A. Baron, "Introduction to the theory of summability of series", Tartu (1966) (In Russian)
[Co]	R.G. Cooke, "Infinite matrices and sequence spaces", Macmillan (1950)
[Ha]	G.H. Hardy, "Divergent series", Clarendon Press (1949)
[Ka]	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