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Fejér kernel

Family of functions in mathematics

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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).

01Definition

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

1) The traditional definition expresses the Fejér kernel $F_{n}(x)$ in terms of the Dirichlet kernel

$F_{n}(x)={\frac {1}{n}}\sum _{k=0}^{n-1}D_{k}(x)$

where

$D_{k}(x)=\sum _{s=-k}^{k}{\rm {e}}^{isx}$

is the $k$ th order Dirichlet kernel.

2) The Fejér kernel $F_{n}(x)$ may also be written in a closed form expression as follows

$F_{n}(x)={\frac {1}{n}}\left({\frac {\sin({\frac {nx}{2}})}{\sin({\frac {x}{2}})}}\right)^{2}={\frac {1}{n}}\left({\frac {1-\cos(nx)}{1-\cos(x)}}\right)$

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:

$D_{k}(x)={\frac {\sin((k+{\frac {1}{2}})x)}{\sin {\frac {x}{2}}}}$ ,

one obtains from the definition of the Fejér kernel above:

F_{n}(x)={\frac {1}{n}}\sum _{k=0}^{n-1}D_{k}(x)={\frac {1}{n}}\sum _{k=0}^{n-1}{\frac {\sin((k+{\frac {1}{2}})x)}{\sin({\frac {x}{2}})}}={\frac {1}{n}}{\frac {1}{\sin({\frac {x}{2}})}}\sum _{k=0}^{n-1}\sin((k+{\frac {1}{2}})x)={\frac {1}{n}}{\frac {1}{\sin ^{2}({\frac {x}{2}})}}\sum _{k=0}^{n-1}{\big [}\sin((k+{\frac {1}{2}})x)\cdot \sin({\frac {x}{2}}){\big ]}

By the trigonometric identity: $\sin(\alpha )\cdot \sin(\beta )={\frac {1}{2}}(\cos(\alpha -\beta )-\cos(\alpha +\beta ))$ , one has

F_{n}(x)={\frac {1}{n}}{\frac {1}{\sin ^{2}({\frac {x}{2}})}}\sum _{k=0}^{n-1}[\sin((k+{\frac {1}{2}})x)\cdot \sin({\frac {x}{2}})]={\frac {1}{n}}{\frac {1}{2\sin ^{2}({\frac {x}{2}})}}\sum _{k=0}^{n-1}[\cos(kx)-\cos((k+1)x)],

which allows evaluation of $F_{n}(x)$ as a telescoping sum:

F_{n}(x)={\frac {1}{n}}{\frac {1}{\sin ^{2}\left({\frac {x}{2}}\right)}}{\frac {1-\cos(nx)}{2}}={\frac {1}{n}}{\frac {1}{\sin ^{2}\left({\frac {x}{2}}\right)}}\sin ^{2}\left({\frac {nx}{2}}\right)={\frac {1}{n}}\left({\frac {\sin({\frac {nx}{2}})}{\sin({\frac {x}{2}})}}\right)^{2}.

3) The Fejér kernel can also be expressed as:

$F_{n}(x)=\sum _{|k|\leq n-1}\left(1-{\frac {|k|}{n}}\right)e^{ikx}$

02Properties

The Fejér kernel is a positive summability kernel. An important property of the Fejér kernel is $F_{n}(x)\geq 0$ with average value of $1$ .

Convolution

The convolution $F_{n}$ is positive: for $f\geq 0$ of period $2\pi$ it satisfies

$0\leq (f*F_{n})(x)={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(y)F_{n}(x-y)\,dy.$

Since

$f*D_{n}=S_{n}(f)=\sum _{|j|\leq n}{\widehat {f}}_{j}e^{ijx},$

we have

$f*F_{n}={\frac {1}{n}}\sum _{k=0}^{n-1}S_{k}(f),$

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

$\|F_{n}*f\|_{L^{p}([-\pi ,\pi ])}\leq \|f\|_{L^{p}([-\pi ,\pi ])}{\text{ for every }}1\leq p\leq \infty \ {\text{for}}\ f\in L^{p}.$

Additionally, if $f\in L^{1}([-\pi ,\pi ])$ , then

$f*F_{n}\rightarrow f$ a.e.

Since $[-\pi ,\pi ]$ is finite, $L^{1}([-\pi ,\pi ])\supset L^{2}([-\pi ,\pi ])\supset \cdots \supset L^{\infty }([-\pi ,\pi ])$ , so the result holds for other $L^{p}$ spaces, $p\geq 1$ 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 $f,g\in L^{1}$ with ${\hat {f}}={\hat {g}}$ , then $f=g$ a.e. This follows from writing

$f*F_{n}=\sum _{|j|\leq n}\left(1-{\frac {|j|}{n}}\right){\hat {f}}_{j}e^{ijt},$

which depends only on the Fourier coefficients.

A second consequence is that if $\lim _{n\to \infty }S_{n}(f)$ exists a.e., then $\lim _{n\to \infty }F_{n}(f)=f$ a.e., since Cesàro means $F_{n}*f$ converge to the original sequence limit if it exists.

Applications

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

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Sources and credits

This article is adapted from the Wikipedia article “Fejér kernel”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Fejér kernel

01Definition

02Properties

Convolution

Applications

Sources and credits

Related topics

Fejér's theorem

Dirichlet kernel

Gibbs phenomenon