Sensitivity analysis for Szegő polynomials (Q841113)

Let \(\mu(t)\) be a distribution function with infinitely many points of increase in the interval \([-\pi,\pi]\) and let \[ \mu_j:=\frac{1}{2\pi} \int^\pi_{-\pi}e^{-ijt}d\mu(t),\;j\in\mathbb{Z}, \] be the moments associated with \(\mu(t)\), where \(\mu (t)\) is scaled so the \(\mu_0=1\). There is an infinite sequence of monic polynomials \(\{\psi_j:j=0,1,\dots\}\), known as Szegö polynomials, that are orthogonal with respect to the inner product \[ \langle f|g\rangle:=\frac{1} {2\pi}\int^\pi_{-\pi}\overline {f(e^{it})}g(e^{it})d\mu(t). \] Numerical methods for weighted least squares approximation by trigonometric polynomials can be conveniently derived and expressed in terms of Szegö polynomials. The present authors discuss the conditioning of several mappings involving Szegö polynomials, with special emphasis on the sensitivity of some approximation problems.