Fejér kernel

In mathematics, the Fejér kernel is a summability kernel used to express the effect of Cesàro summation on Fourier series. It is a non-negative kernel, giving rise to an approximate identity. It is named after the Hungarian mathematician Lipót Fejér (1880–1959).

The Fejér kernel is defined as

where

is the kth order Dirichlet kernel. It can also be written in a closed form as

where this expression is defined.

The Fejér kernel can also be expressed as

The Fejér kernel is a positive summability kernel. An important property of the Fejér kernel is ${\displaystyle F_{n}(x)\geq 0}$ with average value of ${\displaystyle 1}$.

The convolution F_n is positive: for ${\displaystyle f\geq 0}$ of period ${\displaystyle 2\pi }$ it satisfies

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