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A collection of methods for the summation of series of numbers and functions. Introduced by E. Cesàro [[#References|[1]]] and denoted by the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021360/c0213601.png" />.
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{{MSC|40C05}}
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{{TEX|done}}
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A collection of methods for the summation of series of numbers and functions. Introduced by E. Cesàro {{Cite|Ce}} and denoted by the symbol $(C,k)$.
  
 
A series
 
A series
 +
\begin{equation}
 +
\label{eq1}
 +
\sum_{n=0}^\infty a_n
 +
\end{equation}
 +
with partial sums $S_n$ is summable by the Cesàro method of order $k$, or $(C,k)$-summable, with sum $S$ if
 +
$$
 +
\sigma_n^k = \frac{S_n^k}{A_n^k} \rightarrow S, \quad n \rightarrow \infty,
 +
$$
 +
where $S_n^k$ and $A_n^k$ are defined as the coefficients of the expansions
 +
$$
 +
\sum_{n=0}^\infty A_n^k x^n = \frac{1}{(1-x)^{k+1}}, \quad
 +
\sum_{n=0}^\infty S_n^k x^n = \frac{1}{(1-x)^k} \sum_{n=0}^\infty S_n x^n =
 +
\frac{1}{(1-x)^{k+1}}\sum_{n=0}^\infty a_n x^n.
 +
$$
 +
Expressions for $\sigma_n^k$ and $A_n^k$ can be given in the form
 +
$$
 +
\sigma_n^k = \frac{1}{A_n^k}\sum_{\nu=0}^n A_{n-\nu}^{k-1}S_\nu =
 +
\frac{1}{A_n^k}\sum_{\nu=0}^n A_{n-\nu}^{k} a_\nu,
 +
$$
 +
$$
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A_n^k = \binom{k+n}{n} = \frac{(k+1) \cdots (k+n)}{n!}, \quad k \neq,-1,-2,\ldots
 +
$$
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The method $(C,k)$ is a [[Matrix summation method|matrix summation method]] with matrix $[a_{n\nu}]$,
 +
$$
 +
a_{n\nu} =
 +
\begin{cases}
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  \frac{A_{n-\nu}^{k-1}}{A_n^k}, & \nu \leq n, \\
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  0, & \nu > n.
 +
\end{cases}
 +
$$
 +
For $k=0$ the method coincides with ordinary convergence, for $k=1$ it is the method of arithmetic averages. The methods $(C,k)$ are totally regular for $k \geq 0$ and are not regular for $k < 0$. The power of the method increases as $k$ increases: If a series is summable by the method $(C,k)$, then it is summable with the same sum by the method $(C,k')$ for $k' > k > -1$. This property does not hold for $k < -1$. It follows from the summability of the series \ref{eq1} by the method $(C,k)$ that $a_n=o\bigl(n^k\bigr)$. The method $(C,k)$ is equivalent to and compatible with the [[Hölder summation methods|Hölder summation method]] $(H,k)$ and the [[Riesz summation method]] $(R,n,k)$ for $k>0$. For any $k>-1$ the method $(C,k)$ is weaker than the [[Abel summation method]].
  
<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/c/c021/c021360/c0213602.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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Originally, the methods $(C,k)$ were defined by Cesàro for positive integer values of the parameter $k$, and applied to the multiplication of series. They were later extended to arbitrary values of $k$, including complex values. The methods $(C,k)$ have numerous applications, e.g. to the multiplication of series, in the theory of Fourier series, and to other questions.
 
 
with partial sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021360/c0213603.png" /> is summable by the Cesàro method of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021360/c0213605.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021360/c0213607.png" />-summable, with sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021360/c0213608.png" /> if
 
 
 
<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/c/c021/c021360/c0213609.png" /></td> </tr></table>
 
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021360/c02136010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021360/c02136011.png" /> are defined as the coefficients of the expansions
 
 
 
<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/c/c021/c021360/c02136012.png" /></td> </tr></table>
 
 
 
<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/c/c021/c021360/c02136013.png" /></td> </tr></table>
 
 
 
Expressions for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021360/c02136014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021360/c02136015.png" /> can be given in the form
 
 
 
<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/c/c021/c021360/c02136016.png" /></td> </tr></table>
 
 
 
<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/c/c021/c021360/c02136017.png" /></td> </tr></table>
 
 
 
The method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021360/c02136018.png" /> is a [[Matrix summation method|matrix summation method]] with matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021360/c02136019.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/c/c021/c021360/c02136020.png" /></td> </tr></table>
 
 
 
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021360/c02136021.png" /> the method coincides with ordinary convergence, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021360/c02136022.png" /> it is the method of arithmetic averages. The methods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021360/c02136023.png" /> are totally regular for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021360/c02136024.png" /> and are not regular for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021360/c02136025.png" />. The power of the method increases as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021360/c02136026.png" /> increases: If a series is summable by the method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021360/c02136027.png" />, then it is summable with the same sum by the method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021360/c02136028.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021360/c02136029.png" />. This property does not hold for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021360/c02136030.png" />. It follows from the summability of the series (*) by the method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021360/c02136031.png" /> that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021360/c02136032.png" />. The method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021360/c02136033.png" /> is equivalent to and compatible with the summation methods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021360/c02136034.png" /> of Hölder and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021360/c02136035.png" /> of Riesz <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021360/c02136036.png" /> (cf. [[Hölder summation methods|Hölder summation methods]]; [[Riesz summation method|Riesz summation method]]). For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021360/c02136037.png" /> the method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021360/c02136038.png" /> is weaker than Abel's method (cf. [[Abel summation method|Abel summation method]]).
 
  
Originally, the methods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021360/c02136039.png" /> were defined by Cesàro for positive integer values of the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021360/c02136040.png" />, and applied to the multiplication of series. They were later extended to arbitrary values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021360/c02136041.png" />, including complex values. The methods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021360/c02136042.png" /> have numerous applications, e.g. to the multiplication of series, in the theory of Fourier series, and to other questions.
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====References====  
  
====References====
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{|
<table><TR><TD valign="top">[1]</TD> <TD valign="top"E. Cesàro,   ''Bull. Sci. Math.'' , '''14''' : 1 (1890) pp. 114–120</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"G.H. Hardy,   "Divergent series" , Clarendon Press (1949)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"A. Zygmund,   "Trigonometric series" , '''1''' , Cambridge Univ. Press (1988)</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|Ce}}||valign="top"| E. Cesàro, ''Bull. Sci. Math.'', '''14''' : 1 (1890) pp. 114–120
 +
|-
 +
|valign="top"|{{Ref|Ha}}||valign="top"| G.H. Hardy, "Divergent series", Clarendon Press (1949)
 +
|-
 +
|valign="top"|{{Ref|Zy}}||valign="top"| A. Zygmund, "Trigonometric series", '''1''', Cambridge Univ. Press (1988)
 +
|-
 +
|}

Revision as of 14:59, 7 May 2012

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

A collection of methods for the summation of series of numbers and functions. Introduced by E. Cesàro [Ce] and denoted by the symbol $(C,k)$.

A series \begin{equation} \label{eq1} \sum_{n=0}^\infty a_n \end{equation} with partial sums $S_n$ is summable by the Cesàro method of order $k$, or $(C,k)$-summable, with sum $S$ if $$ \sigma_n^k = \frac{S_n^k}{A_n^k} \rightarrow S, \quad n \rightarrow \infty, $$ where $S_n^k$ and $A_n^k$ are defined as the coefficients of the expansions $$ \sum_{n=0}^\infty A_n^k x^n = \frac{1}{(1-x)^{k+1}}, \quad \sum_{n=0}^\infty S_n^k x^n = \frac{1}{(1-x)^k} \sum_{n=0}^\infty S_n x^n = \frac{1}{(1-x)^{k+1}}\sum_{n=0}^\infty a_n x^n. $$ Expressions for $\sigma_n^k$ and $A_n^k$ can be given in the form $$ \sigma_n^k = \frac{1}{A_n^k}\sum_{\nu=0}^n A_{n-\nu}^{k-1}S_\nu = \frac{1}{A_n^k}\sum_{\nu=0}^n A_{n-\nu}^{k} a_\nu, $$ $$ A_n^k = \binom{k+n}{n} = \frac{(k+1) \cdots (k+n)}{n!}, \quad k \neq,-1,-2,\ldots $$ The method $(C,k)$ is a matrix summation method with matrix $[a_{n\nu}]$, $$ a_{n\nu} = \begin{cases} \frac{A_{n-\nu}^{k-1}}{A_n^k}, & \nu \leq n, \\ 0, & \nu > n. \end{cases} $$ For $k=0$ the method coincides with ordinary convergence, for $k=1$ it is the method of arithmetic averages. The methods $(C,k)$ are totally regular for $k \geq 0$ and are not regular for $k < 0$. The power of the method increases as $k$ increases: If a series is summable by the method $(C,k)$, then it is summable with the same sum by the method $(C,k')$ for $k' > k > -1$. This property does not hold for $k < -1$. It follows from the summability of the series \ref{eq1} by the method $(C,k)$ that $a_n=o\bigl(n^k\bigr)$. The method $(C,k)$ is equivalent to and compatible with the Hölder summation method $(H,k)$ and the Riesz summation method $(R,n,k)$ for $k>0$. For any $k>-1$ the method $(C,k)$ is weaker than the Abel summation method.

Originally, the methods $(C,k)$ were defined by Cesàro for positive integer values of the parameter $k$, and applied to the multiplication of series. They were later extended to arbitrary values of $k$, including complex values. The methods $(C,k)$ have numerous applications, e.g. to the multiplication of series, in the theory of Fourier series, and to other questions.

References

[Ba] S.A. Baron, "Introduction to the theory of summability of series", Tartu (1966) (In Russian)
[Ce] E. Cesàro, Bull. Sci. Math., 14 : 1 (1890) pp. 114–120
[Ha] G.H. Hardy, "Divergent series", Clarendon Press (1949)
[Zy] A. Zygmund, "Trigonometric series", 1, Cambridge Univ. Press (1988)
How to Cite This Entry:
Cesàro summation methods. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ces%C3%A0ro_summation_methods&oldid=23234
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article