# 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).

## Definition

The **Fejér kernel** is defined as

where

is the *k*th 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 *F _{n}* is positive: for of period it satisfies

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

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.

- One consequence of the pointwise a.e. convergence is the uniquess of Fourier coefficients: If with , then a.e. This follows from writing , which depends only on the Fourier coefficients.
- A second consequence is that if exists a.e., then a.e., since Cesàro means converge to the original sequence limit if it exists.

## See also

## References

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