Landau kernel

For $k\in\mathbb{N}$ the Landau kernel $L_{k}(t)$ is defined as

L_{k}=\left\{\begin{array}[]{lr}\frac{1}{c_{k}}(1-t^{2})^{k}&\text{if }t\in[-1% ,1]\\ 0&\text{otherwise}\end{array}\right.

with

c_{k}:=\int_{-1}^{1}(1-t^{2})^{k}dt.

$L_{k}$ is nonnegative and continuous on $\mathbb{R}$ . Due to the choice of $c_{k}$ we have

\int_{-\infty}^{\infty}L_{k}(t)dt=1.

Also we have for all positive, real $r$ :

	$\displaystyle\int_{\mathbb{R}\backslash[-r,r]}L_{k}(t)dt$	$\displaystyle\leq\frac{2}{c_{k}}\int_{r}^{1}(1-t^{2})^{k}dt$
		$\displaystyle\leq(k+1)(1-r^{2})^{k}.$

Therefore $(L_{k})_{k\in\mathbb{N}}$ is a Dirac sequence.