Fejér kernel

Short description: Family of functions in mathematics

Plot of several Fejér kernels

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).

## Contents

* 1 Definition
* 2 Properties
* 2.1 Convolution
* 2.2 Applications
* 3 See also
* 4 References

## Definition

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

1) The traditional definition expresses the Fejér kernel Fn(x) in terms of the Dirichlet kernel

Fn(x)=1n∑k=0n−1Dk(x)

where

Dk(x)=∑s=−kkeisx

is the kth order Dirichlet kernel.

2) The Fejér kernel Fn(x) may also be written in a closed form expression as follows[1]

Fn(x)=1n(sin⁡(nx2)sin⁡(x2))2=1n(1−cos⁡(nx)1−cos⁡(x))

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:[2]

Dk(x)=sin⁡((k+12)x)sin⁡x2,

one obtains from the definition of the Fejér kernel above:

Fn(x)=1n∑k=0n−1Dk(x)=1n∑k=0n−1sin⁡((k+12)x)sin⁡(x2)=1n1sin⁡(x2)∑k=0n−1sin⁡((k+12)x)=1n1sin2(x2)∑k=0n−1[sin⁡((k+12)x)⋅sin⁡(x2)]

By the trigonometric identity: sin⁡(α)⋅sin⁡(β)=12(cos⁡(α−β)−cos⁡(α+β)), one has

Fn(x)=1n1sin2(x2)∑k=0n−1[sin⁡((k+12)x)⋅sin⁡(x2)]=1n12sin2(x2)∑k=0n−1[cos⁡(kx)−cos⁡((k+1)x)],

which allows evaluation of Fn(x) as a telescoping sum:

Fn(x)=1n1sin2(x2)1−cos⁡(nx)2=1n1sin2(x2)sin2(nx2)=1n(sin⁡(nx2)sin⁡(x2))2.

3) The Fejér kernel can also be expressed as:

Fn(x)=∑|k|≤n−1(1−|k|n)eikx

## Properties

The Fejér kernel is a positive summability kernel. An important property of the Fejér kernel is Fn(x)≥0 with average value of 1.

### Convolution

The convolution Fn is positive: for f≥0 of period 2π it satisfies

0≤(f*Fn)(x)=12π∫−ππf(y)Fn(x−y)dy.

Since

f*Dn=Sn(f)=∑|j|≤nf^jeijx,

we have

f*Fn=1n∑k=0n−1Sk(f),

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

‖Fn*f‖Lp([−π,π])≤‖f‖Lp([−π,π]) for every 1≤p≤∞ for f∈Lp.

Additionally, if f∈L1([−π,π]), then

f*Fn->f a.e.

Since [−π,π] is finite, L1([−π,π])⊃L2([−π,π])⊃⋯⊃L∞([−π,π]), so the result holds for other Lp spaces, p≥1 as well.

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

* One consequence of the pointwise a.e. convergence is the uniqueness of Fourier coefficients: If f,g∈L1 with f^=g^, then f=g a.e. This follows from writing

f*Fn=∑|j|≤n(1−|j|n)f^jeijt,

which depends only on the Fourier coefficients.

* A second consequence is that if limn->∞Sn(f) exists a.e., then limn->∞Fn(f)=f a.e., since Cesàro means Fn*f 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

1. ↑ Hoffman, Kenneth (1988). Banach Spaces of Analytic Functions. Dover. p. 17. ISBN 0-486-45874-1.
2. ↑ Königsberger, Konrad (in German). Analysis 1 (6th ed.). Springer. pp. 322.

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