From Academic Kids

Cesaro Summability is track twelve of Ănima by the band Tool.

In mathematics, the CesÓro means of a sequence

an

are the terms of the sequence

cn = (a1 + a2 + ... + an)/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

anA

then also

cnA.

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

an = (−1)n

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

CesÓro 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 FejÚr 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 CesÓro means for summing Fourier series, according to the general theory of approximate identities.

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