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One of the methods for summing series and sequences using an infinite matrix. Employing an infinite matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062860/m0628601.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062860/m0628602.png" /> a given sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062860/m0628603.png" /> is transformed into the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062860/m0628604.png" />:
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{{MSC|40C05}}
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062860/m0628605.png" /></td> </tr></table>
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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
 +
$[a_{nk}]$.
  
If the series on the right-hand side converges for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062860/m0628606.png" /> and if the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062860/m0628607.png" /> has a limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062860/m0628608.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062860/m0628609.png" />:
+
A matrix summation method for series can be also defined directly by transforming the series \ref{eq1} into a sequence $(\gamma_n)$:
 
+
\begin{equation}
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062860/m06286010.png" /></td> </tr></table>
+
\label{eq2}
 
+
\gamma_n = \sum_{k=1}^\infty g_{nk}u_k,
then the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062860/m06286011.png" /> is said to be summable by the method determined by the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062860/m06286012.png" />, or simply summable by the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062860/m06286013.png" />, and the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062860/m06286014.png" /> is referred to as its limit in the sense of this summation method. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062860/m06286015.png" /> is regarded as the sequence of partial sums of a series
+
\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
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062860/m06286016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$
 
+
\lim_{n\rightarrow\infty} \gamma_n = s,
then this series is said to be summable to the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062860/m06286017.png" /> by the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062860/m06286018.png" />.
+
$$
 
 
A matrix summation method for series can be also defined directly by transforming the series (1) into a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062860/m06286019.png" />:
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062860/m06286020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
 
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062860/m06286021.png" /> is a given matrix. In this case the series (1) is said to be summable to the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062860/m06286022.png" /> if, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062860/m06286023.png" /> the series on the right-hand side in (2) converges and
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062860/m06286024.png" /></td> </tr></table>
 
 
 
Less often used are matrix summation methods defined by a transformation of a series (1) into a series
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062860/m06286025.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
 
  
 +
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
 
where
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062860/m06286026.png" /></td> </tr></table>
+
\alpha_n = \sum_{k=1}^\infty \alpha_{nk}u_k,
 
+
$$
or by a transformation of a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062860/m06286027.png" /> into a series
+
or by a transformation of a sequence $(s_n)$ into a series
 
+
\begin{equation}
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062860/m06286028.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
\label{eq4}
 
+
\sum_{n=1}^\infty \beta_n,
 +
\end{equation}
 
where
 
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 $s$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062860/m06286029.png" /></td> </tr></table>
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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]]).
 
 
which use matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062860/m06286030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062860/m06286031.png" />, respectively. In these cases the series (1) with the partial sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062860/m06286032.png" /> is summable to the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062860/m06286033.png" /> if the series (3) converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062860/m06286034.png" /> or, respectively, if the series (4) converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062860/m06286035.png" />.
 
  
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|Voronoi summation method]], the [[Cesàro summation methods|Cesàro summation methods]], the [[Euler summation method|Euler summation method]], the [[Riesz summation method|Riesz summation method]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062860/m06286036.png" />, the [[Hausdorff summation method|Hausdorff summation method]], and others (see also [[Summation methods|Summation methods]]).
+
====References====
  
====References====
+
{|
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.H. Hardy,   "Divergent series" , Clarendon Press  (1949)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"R.G. Cooke,   "Infinite matrices and sequence spaces" , Macmillan (1950)</TD></TR><TR><TD valign="top">[3]</TD> <TD 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</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.A. Baron,  "Introduction to the theory of summability of series" , Tartu  (1966)  (In Russian)</TD></TR></table>
+
|-
 +
|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
 +
|-
 +
|}

Latest revision as of 23:33, 8 May 2012

2010 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
How to Cite This Entry:
Matrix summation method. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Matrix_summation_method&oldid=26241
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article