Abstract

Given integers $N geq n geq 0$, we consider the least squares problem of finding the vector of coefficients $vec P$ with respect to a polynomial basis ${ p_0,...,p_n }$, $deg p_j=j$, of a polynomial $P$, $deg P leq n$, which is of best approximation to a given function $f$ with respect to some weighted discrete norm, i.e., which minimizes $sum_{j=0}^N w_n(z_j)^2 |f(z_j) - P(z_j)|^2 $. Here a perturbation of the values $f(z_j)$ leads to some perturbation of the coefficient vector $vec P$. We denote by $kappa_n$ the maximal magnification of relative errors, i.e., the Euclidean condition number of the underlying weighted Vandermonde--like matrix. For the basis of monomials ($p_j(z)=z^j$), the quantity $kappa_n$ equals one when the abscissas are the roots of unity; however, it is known that $kappa_n$ increases exponentially in the case of real abscissas. Here we investigate the $n$th root behavior of $kappa_n$ for some fixed basis and a fixed distribution of (complex) abscissas. An estimate for the $n$th root limit of $kappa_n$ is given in terms of the solution to a weighted constrained energy problem in complex potential theory.