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 (18801959).

Definition

The Fejér kernel has many equivalent definitions. Three such definitions are outlined below:

in terms of the Dirichlet kernel

where

th order Dirichlet kernel.

may also be written in a closed form expression as follows

This closed form expression may be derived from the definitions used above. A proof of this result goes as follows.

Using the fact that the Dirichlet kernel may be written as:

1. ,

one obtains from the definition of the Fejér kernel above:

, one has

as a telescoping sum:

3) The Fejér kernel can also be expressed as:

Properties

.

Convolution

The convolution it satisfies

Since

we have

which is Cesàro summation of Fourier series. By Young's convolution inequality,

, then

1. a.e.

as well.

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

• a.e. This follows from writing

which depends only on the Fourier coefficients.

• converge to the original sequence limit if it exists.

Applications

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

See also

• Fejér's theorem

• Dirichlet kernel

• Gibbs phenomenon

• Charles Jean de la Vallée-Poussin

References