Solving complex approximation problems by semiinfinite - finite optimization techniques: A study on convergence

Summary

We transform a complex approximation problem into an equivalent semiinfinite optimization problem whose constraints are described in terms of a quantity ϕ∈[0,2π[=I. We study the effect of disturbing the problem by replacingI by a compact subsetM⊂I which includes as special case the discrete case whereM consists only of finitely many points. We introduce a measure ɛ for the deviation ofM fromI and show that in any complex approximation problem the minimal distance of the disturbed problem converges quadratically with ɛ→0 to the minimal distance of the undisturbed problem which is a generalization of a result by Streit and Nuttall. We also show that in a linear finite dimensional approximation problem the convergence of the coefficients of the disturbed problem is in general at most linear. There are some graphical representations of best complex approximations computed with the described method.