Strong asymptotics for Sobolev orthogonal polynomials
The authors prove strong asymptotic results for the polynomials \(Q_n\), orthogonal with respect to the inner product \[ \langle f,g,\rangle=\sum_{k=0}^n \int_{\Delta_k} f^{(k)}(x)g^{(k)}(x) d\mu_k(x), \] with \(\{\mu_k\}\) measures supported in \([-1,1]\), satisfying Szegő's condition. The main results concern the following situation: \(\Delta_k\subset\Delta_m\;(0\leq k\leq m-1),\;\Delta_m=[-1,1]\), \(\kappa_n=\langle Q_n,Q_n\rangle\), \(\mathcal F\) the Szegő extremal function corresponding to \(\Delta_m\), \(\varphi(z)\) the conformal map of \([-1,1]\) to the exterior of \(|z|=1/2\). \textbf{A.} If \(d\mu_m(x)=\rho(x)dx\) with \(\rho\) belonging to the Szegő class on \([-1,1]\), \(\{\mu_k\}_0^{m-1}\) finite positive Borel measures, \(\gamma_k(\rho)= \int|P_k(\rho;x)|^2d\mu_m(x)\) with \(P_k\) the orthogonal polynomial of degree \(k\) with respect to \(d\mu_m(x)\), then \[ \lim_{n\rightarrow\infty} {\kappa_n\over n^{2m}\gamma_{n-m}(\rho)}=1, \quad \lim_{n\rightarrow\infty} {Q_n^{(m)}\over n^m\varphi^{n-m}(z)}= {\mathcal F}(z), \quad \lim_{n\rightarrow\infty} {Q_n^{(m)}\over n^mP_{n-m}(\rho;z)}=1, \] the last two limits uniformly on compact subsets of \([-1,1]\). \textbf{B.} If \(\{\mu_k\}_0^m\) are finite, positive, absolutely continuous Borel measures supported on \([-1,1]\), all belonging to the Szegő class, then \[ \lim_{n\rightarrow\infty} {Q_n^{(k)}\over n^k\varphi^{n-k}(z)}= {{\mathcal F}(z)\over (\varphi'(z))^{n-k}}, \qquad \lim_{n\rightarrow\infty} {Q_n^{(k)}\over n^kP_{n-k}(d\mu_m;z)} ={1\over (\varphi'(z))^{n-k}}, \] for \(0\leq k\leq m\), uniformly on compact subsets of \([-1,1]\). An excellent contribution to the steadily growing insight into the nature of Sobolev orthogonal polynomials. The results found, do not ask for information about the location of the zeros of \(Q_n\) (which might be outside the complex hull of the supports of the underlying measures).
scientific article; zbMATH DE number 1369958