The Sensitivity of Least Squares Polynomial Approximation

Given integers N n 0, we consider the least squares problem of nding the vector of coeecients ~ P with respect to a polynomial basis P N j=0 wn(zj) 2 jf(zj) ? P (zj)j 2. Here a perturbation of the values f(zj) leads to some perturbation of the coeecient vector ~ P. We denote by n the maximal magniication of relative errors, i.e., the Euclidean condition number of the underlying weighted Vandermonde{like matrix. For the basis of monomials (pj(z) = z j), the quantity n equals one when the abscissas are the roots of unity; however, it is known that n increases exponentially in the case of real abscissas. Here we investigate the nth{root behavior of n for some xed basis and a xed distribution of (complex) abscissas. An estimate for the nth{root limit of n is given in terms of the solution to a weighted constrained energy problem in complex potential theory.