Landau kernel
LandauKernel
2004-02-25 06:03:43
2006-12-08 08:37:01
Definition
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For $k\in\mathbb{N}$ the \emph{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
$$c_k:=\int_{-1}^1(1-t^2)^kdt.$$
$L_k$ 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 $r$:
\begin{align*}
\int_{\mathbb{R}\backslash[-r,r]}L_k(t)dt&\leq\frac{2}{c_k}\int_r^1(1-t^2)^kdt\\
&\leq(k+1)(1-r^2)^k.
\end{align*}
Therefore $(L_k)_{k\in\mathbb{N}}$ is a Dirac sequence.