# 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.

Title Landau kernel LandauKernel 2013-03-22 14:11:38 2013-03-22 14:11:38 mathwizard (128) mathwizard (128) 7 mathwizard (128) Definition msc 26A30