Fejér kernel

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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).

Plot of several Fejér kernels

Definition

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.[1]

The Fejér kernel can also be expressed as

.

Properties

The Fejér kernel is a positive summability kernel. An important property of the Fejér kernel is with average value of .

Convolution

The convolution Fn is positive: for of period it satisfies

Since , we have , which is Cesàro summation of Fourier series.

By Young's inequality,

for every

for .

Additionally, if , then

a.e.

Since is finite, , so the result holds for other spaces, as well.

If is continuous, then the convergence is uniform, yielding a proof of the Weierstrass theorem.

See also

References

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