Cesàro means

As Cesàro summation or Cesàro averages are to a given sequence of numbers from the first formed sequence elements arithmetic means , respectively. If these converge for growing, one speaks of Cesàro convergence . In the case of series (as a sequence of partial sums) one speaks of Cesàro summability and denotes the limit value as Cesàro sum . This concept formation goes back to the Italian mathematician Ernesto Cesàro and enables an expansion of the normal concept of convergence . It is therefore particularly important in the theory of divergent series and Fourier analysis . ${\displaystyle n}$ ${\displaystyle n}$

definition

For a given sequence of numbers , the arithmetic mean is formed over the first parts of the sequence, i.e. ${\displaystyle (a_{n})_{n\in \mathbb {N} }}$ ${\displaystyle n}$

${\displaystyle {\begin{aligned}\sigma _{1}&=a_{1}\\\sigma _{2}&={\frac {a_{1}+a_{2}}{2}}\\\sigma _{3}&={\frac {a_{1}+a_{2}+a_{3}}{3}}\\&\;\;\vdots \\\sigma _{n}&={\frac {a_{1}+\ldots +a_{n}}{n}}\\&\;\;\vdots \end{aligned}}}$

One then designates the -th Cesàro means or the sequence as the sequence of the Cesàro means . ${\displaystyle \sigma _{n}}$ ${\displaystyle n}$ ${\displaystyle (\sigma _{n})_{n\in \mathbb {N} }}$

The sequence converges to a value , converges to the Cauchy limit theorem the result of Cesàro means against , that is, from the following , respectively . However, the sequence of the Cesàro means can also converge without the output sequence converging. ${\displaystyle (a_{n})_{n\in \mathbb {N} }}$ ${\displaystyle a}$ ${\displaystyle (\sigma _{n})_{n\in \mathbb {N} }}$ ${\displaystyle a}$ ${\displaystyle a_{n}\rightarrow a}$ ${\displaystyle \sigma _{n}\rightarrow a}$ ${\displaystyle {\tfrac {a_{1}+\ldots +a_{n}}{n}}\rightarrow a}$ ${\displaystyle (\sigma _{n})_{n\in \mathbb {N} }}$ ${\displaystyle (a_{n})_{n\in \mathbb {N} }}$

An example of this is the alternating sequence , it is divergent itself, but the sequence of its Cesàro means converges to 0. ${\displaystyle \left((-1)^{n+1}\right)_{n\in \mathbb {N} }=(1,-1,1,-1,\ldots )}$ ${\displaystyle (\sigma _{n})_{n\in \mathbb {N} }=(1,0,{\tfrac {1}{3}},0,{\tfrac {1}{5}},\ldots )=\left({\tfrac {1+(-1)^{n+1}}{2n}}\right)_{n\in \mathbb {N} }}$

This is an extension of the normal concept of convergence for sequences and accordingly designates a sequence as Cesàro-convergent if the sequence of its Cesàro means converges.

Special case rows

An important special case is the application of the Cesàro means or the Cesàro convergence to series , that is to say to the sequence of the partial sums of a series. For a sequence the partial sums of the series are defined as: ${\displaystyle (a_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \textstyle \sum a_{n}}$

${\displaystyle {\begin{aligned}s_{1}&=a_{1}\\s_{2}&=a_{1}+a_{2}\\s_{3}&=a_{1}+a_{2}+a_{3}\\&\;\;\vdots \\s_{n}&=a_{1}+\ldots +a_{n}\\&\;\;\vdots \end{aligned}}}$

The Cesàro means are now formed for this sequence . If these converge, that is , the series Cesàro-convergent , Cesàro-summable or -summizable to the value and writes . The limit value is called the Cesàro sum and the output sequence is then also called Cesàro summable or summable to the value . ${\displaystyle (s_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \sigma _{n}={\tfrac {s_{1}+\ldots +s_{n}}{n}}}$ ${\displaystyle \sigma _{n}\rightarrow \sigma }$ ${\displaystyle \textstyle \sum a_{n}}$ ${\displaystyle C_{1}}$ ${\displaystyle \sigma }$ ${\displaystyle C{\text{-}}\sum _{n=1}^{\infty }a_{n}=\sigma }$ ${\displaystyle \sigma }$ ${\displaystyle (a_{n})_{n\in \mathbb {N} }}$ ${\displaystyle C_{1}}$ ${\displaystyle \sigma }$

If you form the series associated with the above example sequence with , you get the following partial sums: ${\displaystyle (a_{n})_{n\in \mathbb {N} }}$ ${\displaystyle a_{n}=(-1)^{n+1}}$ ${\displaystyle \sum _{n=1}^{\infty }(-1)^{n+1}}$

${\displaystyle {\begin{aligned}s_{1}&=1\\s_{2}&=1-1=0\\s_{3}&=1-1+1=1\\&\;\;\vdots \\s_{2n}&=\sum _{k=1}^{2n}(-1)^{k+1}=0\\s_{2n+1}&=\sum _{k=1}^{2n+1}(-1)^{k+1}=1\\&\;\;\vdots \end{aligned}}}$

The Cesàro means over the sequence of these partial sums are then:

${\displaystyle {\begin{aligned}\sigma _{1}&=1\\\sigma _{2}&={\tfrac {1+0}{2}}={\tfrac {1}{2}}\\\sigma _{3}&={\tfrac {1+0+1}{3}}={\tfrac {2}{3}}\\&\;\;\vdots \\\sigma _{2n}&=\sum _{k=1}^{2n}s_{k}={\tfrac {n}{2n}}={\tfrac {1}{2}}\\\sigma _{2n+1}&=\sum _{k=1}^{2n+1}s_{k}={\tfrac {n+1}{2n+1}}={\tfrac {1}{2}}+{\tfrac {1}{2\cdot (2n+1)}}\\&\;\;\vdots \end{aligned}}}$

It applies and with it . This series is also known as the Grandi series . ${\displaystyle \sigma _{n}\rightarrow {\tfrac {1}{2}}}$ ${\displaystyle C{\text{-}}\sum _{n=1}^{\infty }(-1)^{n+1}={\tfrac {1}{2}}}$

terminology

Many authors define the Cesàro convergence only for series, that is, they only consider the Cesàro means of the associated partial sums. In relation to a series , the terms Cesàro-convergent, Cesàro-summable or Cesàro-summable have the same meaning. But this is not the case when referring to the episode . Here Cesàro convergence and Cesàro summation have a different meaning, because the convergence then relates to the Cesàro means of the sequence terms, while the summation relates to the Cesàro means of the partial sums formed from the sequence terms and thus the summability or convergence of the corresponding series . ${\displaystyle \textstyle \sum a_{n}}$ ${\displaystyle C_{1}}$ ${\displaystyle (a_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \textstyle \sum a_{n}}$

Applications

The application of the Cesàro mean to the Dirichlet kernel in Fourier analysis leads to the Fejér kernel and Fejér's theorem , which describes the convergence behavior of Fourier series . In the theory of divergent series, with the help of the Cesàro mean, certain divergent series can be assigned a limit value in the sense of Cesàro convergence.

literature

• Harro Heuser : Textbook of Analysis - Part 2 . 5th edition, Teubner 1990, ISBN 3-519-42222-0 , p. 155
• Martin Barner , Friedrich Flohr : Analysis I. 3rd edition, Walter de Gruyter 1987, ISBN 311-011517-4 , p. 459
• Guido Walz (Ed.): Lexicon of Mathematics: Volume 1: A to Ei . Springer, 2nd edition, 2017, ISBN 9783662534984 , p. 304
• Douglas N. Clark: Dictionary of Analysis, Calculus, and Differential Equations . CRC Press, 1999, ISBN 9781420049992 , p. 120
• Carl L. DeVito: Harmonic Analysis: A Gentle Introduction . Jones & Bartlett, 2007, ISBN 9780763738938 , p. 43
• Godfrey Harold Hardy : Divergent Series . Clarendon Press, Oxford, 1949, pp. 94-118, particularly p. 96

Web links

• Timo Weidl: The method of arithmetic mean according to Cesaro - part of an Analysis II script (University of Stuttgart)
• Richard Hensh: Infinite Series - Lecture Materials (Michigan State University), pp. 11-15
• Cesaro funds on SOS Math