Landau kernel

The Landau kernel is named after the German number theorist Edmund Landau. The kernel is a summability kernel defined as:¹

$${\displaystyle L_{n}(t)={\begin{cases}{\frac {(1-t^{2})^{n}}{c_{n}}}&{\text{if }}{-1}\leq t\leq 1\\0&{\text{otherwise}}\end{cases}}}$$where the coefficients ${\displaystyle c_{n}}$ are defined as follows:

$${\displaystyle c_{n}=\int _{-1}^{1}(1-t^{2})^{n}\,dt.}$$

Using integration by parts, one can show that:² $${\displaystyle c_{n}={\frac {(n!)^{2}\,2^{2n+1}}{(2n)!(2n+1)}}.}$$ 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, ${\displaystyle L_{n}(t)}$ 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 ${\displaystyle \mathbb {R} }$. These properties are made more concrete in the following section.

The third bullet point means that the area under the graph of the function ${\displaystyle y=K_{n}(t)}$ becomes increasingly concentrated close to the origin as n approaches infinity. This definition lends us to the following theorem.

Proof: We prove the third property only. In order to do so, we introduce the following lemma:

Proof of the Lemma:

Using the definition of the coefficients above, we find that the integrand is even, we may write$${\displaystyle {\frac {c_{n}}{2}}=\int _{0}^{1}(1-t^{2})^{n}\,dt=\int _{0}^{1}(1-t)^{n}(1+t)^{n}\,dt\geq \int _{0}^{1}(1-t)^{n}\,dt={\frac {1}{1+n}}}$$completing the proof of the lemma. A corollary of this lemma is the following: