Landau kernel

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

[math]\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} }[/math]where the coefficients [math]\displaystyle{ c_n }[/math] are defined as follows

[math]\displaystyle{ c_n = \int_{-1}^1 (1-t^2)^n \, dt }[/math]

Visualisation

In these plotted functions, L₁₀(t) represent the blue, L₅₀(t) represent the green, L₁₀₀(t) represent the turquoise, and L₂₀₀(t) represent the purple curve.

Using integration by parts, one can show that: [math]\displaystyle{ c_n = \frac{(n!)^2 \, 2^{2n+1}}{(2n)! (2n+1)}. }[/math] Hence, this implies that the Landau Kernel can be defined as follows: [math]\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} }[/math]

Plotting this function for different values of n reveals that as n goes to infinity, [math]\displaystyle{ L_n(t) }[/math] approaches the Dirac delta function, 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 [math]\displaystyle{ \mathbb{R} }[/math]. These properties are made more concrete in the following section.

Dirac sequences

Definition: Dirac Sequence — A Dirac Sequence is a sequence {[math]\displaystyle{ K_n(t) }[/math]} of functions [math]\displaystyle{ K_n(t) \colon \mathbb{R} \to \mathbb{R} }[/math] that satisfies the following properities:

• [math]\displaystyle{ K_n(t) \geq 0, \, \, \forall t \in \mathbb{R} \text{ and } \forall n \in \mathbb{Z} }[/math]
• [math]\displaystyle{ \int_{-\infty}^{\infty} K_n (t) \, dt =1, \, \forall n }[/math]
• [math]\displaystyle{ \forall \epsilon \gt 0, \, \forall \delta \gt 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 \lt \epsilon }[/math]

The third bullet point means that the area under the graph of the function [math]\displaystyle{ y = K_n(t) }[/math] 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, [math]\displaystyle{ c_n \geq \frac{2}{n+1} }[/math]

Proof of the Lemma:

Using the definition of the coefficients above, we find that the integrand is even, we may write[math]\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} }[/math]completing the proof of the lemma. A corollary of this lemma is the following:

Corollary — For all positive, real [math]\displaystyle{ \delta : }[/math] [math]\displaystyle{ \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 }[/math]

See also

• Poisson Kernel
• Fejer Kernel
• Dirichlet Kernel