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 has many equivalent definitions. Three such definitions are outlined below:
1) The traditional definition expresses the Fejér kernel <math>F_n(x)</math> in terms of the Dirichlet kernel
- <math> F_n(x) = \frac{1}{n} \sum_{k=0}^{n-1}D_k(x) </math>
where
- <math>D_k(x)=\sum_{s=-k}^k {\rm e}^{isx}</math>
is the <math>k</math>th order Dirichlet kernel.
2) The Fejér kernel <math>F_n(x)</math> may also be written in a closed form expression as follows<ref>Template:Cite book</ref>
<math> 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)</math>
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:<ref>Template:Cite book</ref>
- <math>D_k(x)=\frac{
\sin((k+\frac{1}{2})x)}{\sin\frac{x}{2}}</math>, one obtains from the definition of the Fejér kernel above:
- <math display=block>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] </math>
By the trigonometric identity: <math>\sin(\alpha)\cdot\sin(\beta)=\frac{1}{2}(\cos(\alpha-\beta)-\cos(\alpha+\beta))</math>, one has
- <math display=block>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)],</math> which allows evaluation of <math>F_n(x)</math> as a telescoping sum:
- <math display=block>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.</math>
3) The Fejér kernel can also be expressed as:
- <math> F_n(x)=\sum_{ |k| \leq n-1} \left(1-\frac{ |k| }{n}\right)e^{ikx} </math>
Properties
The Fejér kernel is a positive summability kernel. An important property of the Fejér kernel is <math>F_n(x) \ge 0</math> with average value of <math>1 </math>.
Convolution
The convolution <math>F_n</math> is positive: for <math>f \ge 0</math> of period <math>2 \pi</math> it satisfies
- <math>0 \le (f*F_n)(x)=\frac{1}{2\pi}\int_{-\pi}^\pi f(y) F_n(x-y)\,dy.</math>
Since
- <math>f*D_n=S_n(f)=\sum_{|j|\le n}\widehat{f}_je^{ijx} ,</math>
we have
- <math>f*F_n=\frac{1}{n}\sum_{k=0}^{n-1}S_k(f),</math>
which is Cesàro summation of Fourier series. By Young's convolution inequality,
- <math>\|F_n*f \|_{L^p([-\pi, \pi])} \le \|f\|_{L^p([-\pi, \pi])} \text{ for every } 1 \le p \le \infty\ \text{for}\ f\in L^p.</math>
Additionally, if <math>f\in L^1([-\pi,\pi])</math>, then
- <math>f*F_n \rightarrow f</math> a.e.
Since <math>[-\pi,\pi]</math> is finite, <math>L^1([-\pi,\pi])\supset L^2([-\pi,\pi])\supset\cdots\supset L^\infty([-\pi,\pi])</math>, so the result holds for other <math>L^p</math> spaces, <math>p\ge1</math> as well.
If <math>f</math> 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 <math>f,g\in L^1</math> with <math>\hat{f}=\hat{g}</math>, then <math>f=g</math> a.e. This follows from writing
- <math>f*F_n=\sum_{|j|\le n}\left(1-\frac{|j|}{n}\right)\hat{f}_je^{ijt},</math>
which depends only on the Fourier coefficients.
- A second consequence is that if <math>\lim_{n\to\infty}S_n(f)</math> exists a.e., then <math>\lim_{n\to\infty}F_n(f)=f</math> a.e., since Cesàro means <math>F_n*f</math> converge to the original sequence limit if it exists.
Applications
The Fejér kernel is used in signal processing and Fourier analysis.
See also
References
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