The Landau kernel is named after the German number theorist Edmund Landau. The kernel is a summability kernel defined as:
L n ( t ) = { ( 1 − t 2 ) n c n if − 1 ≤ t ≤ 1 0 otherwise {\displaystyle L_{n}(t)={\begin{cases}{\frac {(1-t^{2})^{n}}{c_{n}}}&{\text{if }}{-1}\leq t\leq 1\\0&{\text{otherwise}}\end{cases}}} where the coefficients c n {\displaystyle c_{n}} are defined as follows:
c n = ∫ − 1 1 ( 1 − t 2 ) n d t . {\displaystyle c_{n}=\int _{-1}^{1}(1-t^{2})^{n}\,dt.}
Visualisation Using integration by parts, one can show that:
c n = ( n ! ) 2 2 2 n + 1 ( 2 n ) ! ( 2 n + 1 ) . {\displaystyle c_{n}={\frac {(n!)^{2}\,2^{2n+1}}{(2n)!(2n+1)}}.}
Hence, this implies that the Landau kernel can be defined as follows: L n ( t ) = { ( 1 − t 2 ) n ( 2 n ) ! ( 2 n + 1 ) ( n ! ) 2 2 2 n + 1 for t ∈ [ − 1 , 1 ] 0 elsewhere {\displaystyle L_{n}(t)={\begin{cases}(1-t^{2})^{n}{\frac {(2n)!(2n+1)}{(n!)^{2}\,2^{2n+1}}}&{\text{for }}t\in [-1,1]\\0&{\text{elsewhere}}\end{cases}}}
Plotting this function for different values of n reveals that as n goes to infinity, L n ( t ) {\displaystyle L_{n}(t)} approaches the Dirac delta function as a distribution, as seen in the image, where the following functions are plotted.
Properties Some general properties of the Landau kernel is that it is nonnegative and continuous on R {\displaystyle \mathbb {R} } . These properties are made more concrete in the following section.
Dirac sequences
The third bullet point means that the area under the graph of the function y = K n ( t ) {\displaystyle y=K_{n}(t)} becomes increasingly concentrated close to the origin as n approaches infinity. This definition lends us to the following theorem.
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