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 has many equivalent definitions.

We outline three such definitions below: 1) The traditional definition expresses the Fejér kernel

in terms of the Dirichlet kernel:

where is the kth order Dirichlet kernel.

2) The Fejér kernel

may also be written in a closed form expression as follows[1]

sin ⁡ (

sin ⁡ (

This closed form expression may be derived from the definitions used above.

The proof of this result goes as follows.

First, we use the fact that the Dirichlet kernel may be written as:[2] Hence, using the definition of the Fejér kernel above we get: Using the trigonometric identity:

sin ⁡ ( α ) ⋅ sin ⁡ ( β ) =

( cos ⁡ ( α − β ) − cos ⁡ ( α + β ) )

Hence it follows that: 3) The Fejér kernel can also be expressed as:

The Fejér kernel is a positive summability kernel.

An important property of the Fejér kernel is

The convolution Fn is positive: for

2 π

, which is Cesàro summation of Fourier series.

By Young's convolution inequality, Additionally, if

( [ − π , π ] )

[ − π , π ]

, so the result holds for other

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

The Fejér kernel is used in signal processing and Fourier analysis.