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Cesàro mean

 

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 series \sum_{n=1}^\infty (-1)^n which is Cesàro summable to 1/2, meaning the Cesàro means of the partial sums have limit 1/2. (The partial sums are 1,0,1,0,1,0,... like the previous sequence.)

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.

See also

References

  1. ^ a b Hardy, G. H. (1992). Divergent Series. Providence: American Mathematical Society.  
  2. ^ Katznelson, Yitzhak (1976). An Introduction to Harmonic Analysis. New York: Dover Publications.  

External links

  • Cesaro mean at PlanetMath
  • Cesaro mean at SOS Math
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