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Landau kernel
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(Definition)
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For $k\in\mathbb{N}$ the Landau kernel $L_k(t)$ is defined as
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 $r$ :
Therefore $(L_k)_{k\in\mathbb{N}}$ is a Dirac sequence.
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"Landau kernel" is owned by mathwizard.
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(view preamble | get metadata)
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 1624 times total.
Classification:
| AMS MSC: | 26A30 (Real functions :: Functions of one variable :: Singular functions, Cantor functions, functions with other special properties) |
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Pending Errata and Addenda
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![$\displaystyle L_k=\left\{\begin{array}{lr} \frac{1}{c_k}(1-t^2)^k&\text{if }t\in[-1,1]\ 0&\text{otherwise} \end{array}\right.$](http://images.planetmath.org:8080/cache/objects/5624/js/img1.png "$\displaystyle L_k=\left\{\begin{array}{lr} \frac{1}{c_k}(1-t^2)^k&\text{if }t\in[-1,1]\ 0&\text{otherwise} \end{array}\right.$")
![$\displaystyle \int_{\mathbb{R}\backslash[-r,r]}L_k(t)dt$](http://images.planetmath.org:8080/cache/objects/5624/js/img2.png "$\displaystyle \int_{\mathbb{R}\backslash[-r,r]}L_k(t)dt$")
