The method IRLS for some best \(\ell_{p}\) norm solutions
of under- or overdetermined linear systems

Abstract. The iteratively reweighted least sqaures (IRLS) method is not recommended for the cases of \(p=1,\infty\) among the many approaches to solve such problems. However, some useful properties are pointed out here: It is simple to use, it offers fast convergence for polynomial best uniform approximations and it can be easily extended to multivariate polynomial (or more generally: linear) approximations. It helps finding the extremal set, moreover, it has offered invaluable help to formulate a generalized de la Vallée Poussin type characterization \((p=\infty)\), when Haar condition does not apply.

Key words and phrases. Best solutions of under- or overdetermined linear systems, pseudoinverse, Haar condition, weighted least squares, uniform approximation of functions in one or more variables, generalized de la Vallée Poussin characterization.