Cesàro mean

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In mathematics, the Cesàro means (also called Cesàro averages) of a sequence (),  are the arithmetic means  
()  of the terms of sequence (). [1]:96  This concept is named after Ernesto Cesàro (1859 - 1906).

A basic result [1]:100-102 states that the limit of a convergent sequence equals the limit of its Cesàro mean.

That is, the operation of taking Cesàro means preserves the convergence and the limit of a sequence. This is the basis for using Cesàro means in a summability method in the theory of divergent series.

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.

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References[edit]

  1. ^ a b Hardy, G. H. (1992). Divergent Series. Providence: American Mathematical Society. ISBN 978-0-8218-2649-2. 
  2. ^ Katznelson, Yitzhak (1976). An Introduction to Harmonic Analysis. New York: Dover Publications. ISBN 978-0-486-63331-2. 

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