Fejér kernel

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In mathematics, the Fejér kernel is used to express the effect of Cesàro summation on Fourier series. It is a non-negative kernel, giving rise to an approximate identity.

The Fejér kernel is defined as

F_n(x) = \frac{1}{n} \sum_{k=0}^{n-1}D_k(x),

where Dk(x) is the kth order Dirichlet kernel. It can also be written in a closed form as

F_n(x) = \frac{1}{n} \left(\frac{\sin \frac{n x}{2}}{\sin \frac{x}{2}}\right)^2,

where this expression is defined. It is named after the Hungarian mathematician Lipót Fejér (1880–1959).


Plot of several Fejér kernels
Plot of several Fejér kernels

[edit] See also