Bernstein-Szegő's theorem for Sobolev orthogonal polynomials
Let \(\Gamma\) be a rectifiable curve or arc in \(\mathbb{C}\) (not reducing to a single point) for which the arclength parametrization \(z(s)\) satisfies \[ z\in C^2,\;|z''(x)-z''(y)|\leq M|x-y|^{\alpha} (M,\alpha>0\text{ fixed}; x,y\in\Gamma). \] Consider two measures \(\mu_0, \mu_1\) supported on \(\Gamma\) and define the inner products \[ \langle f,g\rangle_i =\int f{\overline g} d\mu_i\;(i=0,1),\quad (f,g)_S= \langle f,g\rangle_0 + \langle f,g\rangle_1 \] (the last one called Sobolev inner product). For an arbitrary Borel measure \(\mu\) with infinitely many points of increase the \(n\)-th monic orthogonal polynomial is denoted by \(P_n(\mu;z)\) and \[ \tau_n(\mu)=\int |P_n(\mu;z)|d\mu(z). \] Finally put \(k_n=(Q_n,Q_n)_S=\min\{(Q,Q)_S\mid Q\text{ is monic}\}\). The author then proves the results \[ \lim_{n\rightarrow\infty} {k_n\over n^2\tau_{n-1}(\mu)}=1, \tag{1} \] where \(\rho_1=\mu_1^{'}\) is in the Szegő class on \(\Gamma\), \(\mu_0\) any Borel measure on \(\Gamma\). \[ \lim_{n\rightarrow\infty} {Q_n'(z)\over nP_{n-1}(\mu_1;z)}=1, \tag{2} \] uniformly on compact subsets of \(\Omega\) (the open connected component of \({\overline \mathbb{C}}\setminus\Gamma\) containing infinity). \[ \lim_{n\rightarrow\infty} {Q_n(z)\over nP_n(\mu_1;z)}={1\over \Phi'(z)}, \tag{3} \] if \(\rho_0=\mu_0'\) is in the Szegő class, \(\Phi\) the conformal mapping of \(\Omega\) onto the exterior of a circle centered at \(z=0\). Moreover, equivalent formulations of (2) and (3) using the Szegő function of \(\rho_1=\mu_1^{'}\) are given. A quite interesting paper.
scientific article; zbMATH DE number 1419734