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.