PlanetMath: Landau kernel

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Landau kernel

(Definition)

For $k\in\mathbb{N}$ the Landau kernel is defined as

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

with

$\displaystyle c_k:=\int_{-1}^1(1-t^2)^kdt.$

is nonnegative and continuous on $\mathbb{R}$ . Due to the choice of

we have

$\displaystyle \int_{-\infty}^\infty L_k(t)dt=1.$

Also we have for all positive, real

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

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

"Landau kernel" is owned by mathwizard.

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Cross-references: Dirac sequence, real, positive, continuous

This is version 4 of Landau kernel, born on 2004-02-25, modified 2006-12-08.
Object id is 5624, canonical name is LandauKernel.
Accessed 1196 times total.

Classification:

26A30 (Real functions :: Functions of one variable :: Singular functions, Cantor functions, functions with other special properties)

Pending Errata and Addenda

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