Landau kernel

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

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

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

Visualisation

Using integration by parts, one can show that:^[2] $c_{n}={\frac {(n!)^{2}\,2^{2n+1}}{(2n)!(2n+1)}}.$ Hence, this implies that the Landau Kernel can be defined as follows: $L_{n}(t)={\begin{cases}(1-t^{2})^{n}{\frac {(2n)!(2n+1)}{(n!)^{2}\,2^{2n+1}}}&{\text{for t}}\in [-1,1]\\0&{\text{elsewhere}}\end{cases}}$

Plotting this function for different values of n reveals that as n goes to infinity, $L_{n}(t)$ approaches the Dirac delta function, as seen in the image,^[1] where the following functions are plotted.

Properties

Some general properties of the Landau kernel is that it is nonnegative and continuous on $\mathbb {R}$ . These properties are made more concrete in the following section.

Dirac sequences

Definition: Dirac Sequence — A Dirac Sequence is a sequence { $K_{n}(t)$ } of functions $K_{n}(t)\colon \mathbb {R} \to \mathbb {R}$ that satisfies the following properities:

$K_{n}(t)\geq 0,\,\,\forall t\in \mathbb {R} {\text{ and }}\forall n\in \mathbb {Z}$
$\int _{-\infty }^{\infty }K_{n}(t)\,dt=1,\,\forall n$
$\forall \epsilon >0,\,\forall \delta >0,\,\exists N\in \mathbb {Z} _{+}{\text{ such that }}\forall n\geq N:\int _{\mathbb {R} \setminus [-\delta ,\delta ]}K_{n}(t)\,dt=\int _{-\infty }^{-\delta }K_{n}(t)\,dt+\int _{\delta }^{\infty }K_{n}(t)\,dt<\epsilon$

The third bullet point means that the area under the graph of the function $y=K_{n}(t)$ 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, $c_{n}\geq {\frac {2}{n+1}}$

Proof of the Lemma:

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

Corollary — For all positive, real $\delta :$ $\int _{\mathbb {R} \setminus [-\delta ,\delta ]}K_{n}(t)\,dt\leq {\frac {2}{c_{n}}}\int _{\delta }^{1}(1-t^{2})^{n}\,dt\leq (n+1)(1-r^{2})^{n}$