On interpolation and approximation problems in numerical linear algebra
Olivier Sète · DepositOnce · 2016
This doctoral thesis is on interpolation and approximation problems in the complex plane which are motivated by questions in numerical linear algebra. In the first part of this thesis, we consider the zeros of rational harmonic functions. In this context we sharpen a bound on the number of zeros of such functions, and show that extremal functions, i.e., rational harmonic functions attaining this bound, are always regular. Moreover, we analyze the change of the number of zeros of rational harmonic functions when adding a pole. This generalizes a construction of Rhie (ArXiv Astrophysics e-prints, 2003), who gave the first examples of extremal functions. Her examples, however, have high rotational symmetry. Our analysis yields in particular a construction principle for general non-symmetric extremal functions. We apply this result in the context of gravitational microlensing in astrophysics, to obtain a construction principle for unsymmetric gravitational point lenses for which maximal lensing occurs. The second part of this thesis is on approximation of analytic functions by series of Faber-Walsh polynomials, which generalize Faber olynomials to compact sets with several components. The Faber-Walsh polynomials are defined through conformal maps of multiply connected domains onto lemniscatic domains, which generalize the Riemann mapping. We first construct two analytic examples of such maps, and give a general construction principle for these maps for certain polynomial pre-images. With these results we derive general properties of the Faber-Walsh polynomials, and relate them to the classical Faber and Chebyshev polynomials. We further present examples of Faber-Walsh polynomials for two real intervals, and also for two nonreal sets consisting of several components.