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. Three such definitions are outlined below:

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

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

where

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

is the ${\displaystyle k}$th order Dirichlet kernel.

2) The Fejér kernel ${\displaystyle F_{n}(x)}$ may also be written in a closed form expression as follows¹

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

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:²

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

one obtains from the definition of the Fejér kernel above:

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

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

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

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

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

3) The Fejér kernel can also be expressed as:

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

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

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

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

Since

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

we have

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

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

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

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

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

Since ${\displaystyle [-\pi ,\pi ]}$ is finite, ${\displaystyle L^{1}([-\pi ,\pi ])\supset L^{2}([-\pi ,\pi ])\supset \cdots \supset L^{\infty }([-\pi ,\pi ])}$, so the result holds for other ${\displaystyle L^{p}}$ spaces, ${\displaystyle p\geq 1}$ as well.

If ${\displaystyle f}$ 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 ${\displaystyle f,g\in L^{1}}$ with ${\displaystyle {\hat {f}}={\hat {g}}}$, then ${\displaystyle f=g}$ a.e. This follows from writing

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

which depends only on the Fourier coefficients.

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

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