Landau kernel

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

$$L_n(t)=\{\begin{array}{cc}\frac{(1-t^2{)}^n}{c_n}& \text{if }-1\le t\le 1\\ 0& \text{otherwise}\end{array}$$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: $$c_n=\frac{(n!{)}^2{2}^{2n+1}}{(2n)!(2n+1)}.$$ Hence, this implies that the Landau kernel can be defined as follows: $$L_n(t)=\{\begin{array}{cc}(1-t^2{)}^n\frac{(2n)!(2n+1)}{(n!{)}^2{2}^{2n+1}}& \text{for }t\in [-1,1]\\ 0& \text{elsewhere}\end{array}$$

Plotting this function for different values of n reveals that as n goes to infinity, $L_n(t)$ 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 $ℝ$. These properties are made more concrete in the following section.

Dirac sequences

Template:Math theorem 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. Template:Math theorem Proof: We prove the third property only. In order to do so, we introduce the following lemma: Template:Math theorem Proof of the Lemma:

Using the definition of the coefficients above, we find that the integrand is even, we may write$$\frac{c_n}{2}={\int }_0^1(1-t^2{)}^{n}dt={\int }_0^1(1-t{)}^n(1+t{)}^{n}dt\ge {\int }_0^1(1-t{)}^{n}dt=\frac{1}{1+n}$$completing the proof of the lemma. A corollary of this lemma is the following: Template:Math theorem

See also

• Poisson kernel
• Fejér kernel
• Dirichlet kernel