The Landau kernel is named after the German number theorist Edmund Landau. The kernel is a summability kernel defined as:[1]
where the coefficients
are defined as follows
Visualisation
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.
Properties
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.
Dirac sequences
Definition: Dirac Sequence — A Dirac Sequence is a sequence {
} of functions
that satisfies the following properities:
![{\displaystyle K_{n}(t)\geq 0,\,\,\forall t\in \mathbb {R} {\text{ and }}\forall n\in \mathbb {Z} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/827ebe072bee14f3755771d323b4851049c4a4e9)
![{\displaystyle \int _{-\infty }^{\infty }K_{n}(t)\,dt=1,\,\forall n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a83d5ff1323d9977d47a4b0f3b4bcd7c2b0f2379)
![{\displaystyle \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 }](https://wikimedia.org/api/rest_v1/media/math/render/svg/358470daa11d36a46f9ceb0fe7acbeb4926d0c7b)
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 write
completing the proof of the lemma. A corollary of this lemma is the following:
Corollary — For all positive, real
See also
References
- ^ a b Terras, Audrey (May 25, 2009). "Lecture 8. Dirac and Weierstrass" (PDF).
- ^ Hilber, Courant. Methods of Mathematical Physics, Vol. I. p. 84.