Landau kernel

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

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

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:

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

  1. Terras, Audrey (May 25, 2009). "Lecture 8. Dirac and Weierstrass" (PDF).
  2. Hilber, Courant. Methods of Mathematical Physics, Vol. I. p. 84.
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