Cesàro mean Guide, Meaning , Facts, Information and Description

In mathematics, the Cesàro means of a sequence

a_n

are the terms of the sequence

c_n = (a₁ + a₂ + ... + a_n)/n

constructed as the arithmetic mean of the first n elements. This concept is named after Ernesto Cesàro; (1859 - 1906).

A basic result states that if

a_n → A

then also

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. There are certainly many examples for which the Cesàro means converge, but the original sequence does not: for example with

a_n = (−1)ⁿ

we have an oscillating sequence, but the means have limit 0.

Cesàro means are often applied to Fourier series, 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 Dirichet 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.

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