The Landau kernel is named after the German number theorist Edmund Landau.
The kernel is a summability kernel defined as:[1]
n
2
n
{\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
are defined as follows:
{\displaystyle c_{n}=\int _{-1}^{1}(1-t^{2})^{n}\,dt.}
Using integration by parts, one can show that:[2]
Hence, this implies that the Landau kernel can be defined as follows:
{\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,
approaches the Dirac delta function, as seen in the image,[1] where the following functions are plotted.
Some general properties of the Landau kernel is that it is nonnegative and continuous on
These properties are made more concrete in the following section.
Definition: Dirac sequence — A Dirac sequence is a sequence
of functions
that satisfies the following properities: The third bullet point means that the area under the graph of the function
becomes increasingly concentrated close to the origin as n approaches infinity.
This definition lends us to the following theorem.
Theorem — The sequence of Landau kernels is a Dirac sequence Proof: We prove the third property only.
In order to do so, we introduce the following lemma: Lemma — The coefficients satsify the following relationship,
Proof of the Lemma: Using the definition of the coefficients above, we find that the integrand is even, we may write
completing the proof of the lemma.
A corollary of this lemma is the following: Corollary — For all positive, real
δ :
∖ [ − δ , δ ]
δ