# Cesàro mean

In mathematics, the Cesàro means (also called Cesàro averages) of a sequence (an) are the terms of the sequence (cn), where

$c_n = \frac{1}{n} \sum_{i=1}^{n} a_i$

is the arithmetic mean of the first n elements of (an). [1]:96 This concept is named after Ernesto Cesàro (1859 - 1906).

A basic result [1]:100-102 states that if

$\lim_{n \to \infty} a_n = A$

then also

$\lim_{n \to \infty} c_n = A.$

That is, the operation of taking Cesàro means preserves convergent sequences and their limits. This is the basis for taking Cesàro means as a summability method in the theory of divergent series. If the sequence of the Cesàro means is convergent, the series is said to be Cesàro summable. There are certainly many examples for which the sequence of Cesàro means converges, but the original sequence does not: for example with

$a_n=\begin{cases}1&\mbox{if }n=2k-1, \\ 0&\mbox{if }n=2k\end{cases}$,

we have an oscillating sequence, but the means have limit $\frac{1}{2}$. (See also Grandi's series.)

Another example is the sequence $a_n=(-1)^n$ which is Cesàro summable to $1/2$ and has Cesàro-mean $0$.

Cesàro means are often applied to Fourier series, [2]:11-13 since the means (applied to the trigonometric polynomials making up the symmetric partial sums) are more powerful in summing such series than pointwise convergence. The kernel that corresponds is the Fejér kernel, replacing the Dirichlet kernel; it is positive, while the Dirichlet kernel takes both positive and negative values. This accounts for the superior properties of Cesàro means for summing Fourier series, according to the general theory of approximate identities.

A generalization of the Cesàro mean is the Stolz-Cesàro theorem.

The Riesz mean was introduced by M. Riesz as a more powerful but substantially similar summability method.