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Cesàro Mean


The Cesàro means of a function f are the arithmetic means

 sigma_n=1/n(s_0+...+s_(n-1)),
(1)

n=1, 2, ..., where the addend s_k is the kth partial sum

 s_k(x)=sum_(j=-k)^kc_je^(ijx)
(2)

of the Fourier series

 sum_(n=-infty)^inftyc_ne^(inx)
(3)

for f. Here, c_j is the jth coefficient

 c_j=1/(2pi)int_(-pi)^pif(x)e^(-ijx)dx
(4)

in the Fourier expansion for f, j=0,+/-1,+/-2,....

Cesàro means are of particular importance in the study of function spaces. For example, a well-known fact is that if f is a p-integrable function for 1<=p<infty, the Cesàro means of f converge to f in the L^p-norm and, moreover, if f is continuous, the convergence is uniform. The nth Cesàro mean of f can also be obtained by integrating f against the nth Fejer kernel.


See also

Fejér's Integral, Fourier Series

This entry contributed by Christopher Stover

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References

Hoffman, K. Banach Spaces of Analytic Functions. New York: Dover Publications Inc., 2007.

Cite this as:

Stover, Christopher. "Cesàro Mean." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/CesaroMean.html

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