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Landau kernel

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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

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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

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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

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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:

Corollary — For all positive, real

See also

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References

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