Using integration by parts, one can show that:[2]
Hence, this implies that the Landau Kernel can be defined as follows:
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 writecompleting the proof of the lemma. A corollary of this lemma is the following: