The Cesàro means of a function are the arithmetic means
(1)
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, 2, ..., where the addend is the th partial sum
(2)
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of the Fourier series
(3)
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for . Here, is the th coefficient
(4)
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in the Fourier expansion for , .
Cesàro means are of particular importance in the study of function spaces. For example, a well-known fact is that if is a -integrable function for , the Cesàro means of converge to in the -norm and, moreover, if is continuous, the convergence is uniform. The th Cesàro mean of can also be obtained by integrating against the th Fejer kernel.