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An undergraduate miscellany: from the Laplace boundary problem to the sampling phenomenon in Colombeau's algebra

In this talk I will give a brief overview of two different lines of my undergraduate research. The first part of talk concerns the extended potential function that enables computing the solution of the continuous interior Dirichlet problem on basis of discrete problems solutions. This research deals with reduction of computational efforts needed for solving the Laplace boundary problem by finite-difference approach. A novel technique of prediction by polynomial extrapolation will be presented. In the second part of the talk I will discuss the application of distribution calculus in signal and system analysis. In many practical situations a product of an arbitrary function and Dirac's delta distribution frequently arises. According to Schwartz's "impossibility result" this product cannot be consistently defined within the Schwartz vector space. This deficiency of classical approach can be moderated by introducing Colombeau's algebra of generalized functions. I will present the application of Colombeau's algebra and a concept of weak equality in deriving a key formula that relates the Laplace transform of sampled and original function seen as rapidly decreasing generalized function of bounded rate. The obtained results do not suppress the functional nature of Dirac's delta distribution, and therefore they differ from those classically acquired. In order to establish a relationship between new and classical results a delayed sampling procedure is introduced.

Type of Seminar:
IfA Seminar
Marko Seslija
May 14, 2008   17.15

ETL K 25
Contact Person:

Melanie Zeilinger
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