Difference between revisions of "Cesàro summation methods"
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| − | A collection of methods for the summation of series of numbers and functions. Introduced by E. Cesàro | + | {{MSC|40C05}} |
| + | {{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 | ||
| + | $$ | ||
| + | \sigma_n^k = \frac{1}{A_n^k}\sum_{\nu=0}^n A_{n-\nu}^{k-1}S_\nu = | ||
| + | \frac{1}{A_n^k}\sum_{\nu=0}^n A_{n-\nu}^{k} a_\nu, | ||
| + | $$ | ||
| + | $$ | ||
| + | A_n^k = \binom{k+n}{n} = \frac{(k+1) \cdots (k+n)}{n!}, \quad k \neq,-1,-2,\ldots | ||
| + | $$ | ||
| + | The method $(C,k)$ is a [[Matrix summation method|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 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]]. | ||
| − | + | 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. | |
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| − | + | ====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|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=26191
Cesàro summation methods. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ces%C3%A0ro_summation_methods&oldid=26191
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article