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Chebychev Approximation by Polynomials, Rational Functions and Splines
Intervenant(s) : Guido Walz (Wilhelm Büchner Hochschule) We give a survey of the theory of best approximation in the sense of Chebychev, i.e., using the uniform norm (Chebychev norm). If the approximating set is a Haar subspace, there are existence- and uniqueness-Theorems as well as a full characterization of the best approximation in terms of alternants. For the numerical computation of the best approximation there exist well-established algorithms by Remez, either with single or with simultaneous exchange of the reference points.
In the second part we look onto Chebychev approximation by so-called weak Haar subspaces, the prototype of which is given by spline spaces, and give similar results as above.
Depending on the time and the interests of the audience, the talk is concluded by some results on nonlinear Chebychev approximation, in particular approximation by rational functions.
Marc.Mezzarobba (at) null