Asymptotic behavior of Sobolev-type orthogonal polynomials on the unit circle
The authors consider the polynomials \(\psi_n(Z)\) \((n=0,1,\dots)\) which are orthogonal with respect to the Sobolev inner product on the unit circle, namely: \[ \langle f,g\rangle= \int^{2\pi}_0 f(e^{i\theta}) \overline {g (e^{i\theta})} d\mu(\theta+ f(\mathbb{Z}) Ag(\mathbb{Z})^H. \] Here \(\mu\) is supposed to be a probability measure whose support consists of an infinite set of points contained in \([0,2\pi]\), \(A\) is a given \(M\times M\) positive semi-definite matrix while \(\mathbb{Z}\) refers to a set of given points \(z_1,z_2, \dots,z_m\) such that \[ F(\mathbb{Z})= \bigl(f(z_1), \dots,f^{(\ell_1)} (z_1),\dots, f(z_m),\dots, f^{ (\ell_m)} (z_m) \bigr). \] Here \(\ell_1,\ell_2,\dots,\ell_m\) are also given nonnegative integers such that \(M=\ell_1+\ell_2+\cdots+\ell_m+m\). The superindex \(H\) appearing in \(g^H\) refers to the conjugate transpose of the vector \(g\). Their main aim is to compute the limits of the functions \[ \begin{aligned} \psi_n^{(k)} (z)/ \varphi_n^{(k)} (z), & |z|>1,k=0,1, \dots\\ \psi_n(z)/ \varphi_n^*(z), & |z|< 1\end{aligned} \] as well as the ratio \(k_n/ \gamma_n\) when \(n\to\infty\). Here \(\varphi_n (n=0,1, \dots)\) denotes the sequence of orthonormal polynomials with respect to the measure \(\mu\) while \(\varphi_n^*(z)\) is defined by \[ \varphi_n^* (z)=z^n \overline {\varphi_n (1/ \overline z)}. \] Finally, the positive numbers \(\gamma_n\) and \(k_n\) refer to the coefficients of the leading terms in the polynomials \(\psi_n(z)= \gamma_nz^n +\dots\) and \(\varphi_n(z)= k_nz^n+ \dots\). To get the result they prove first four lemmas and two theorems depending all on matrix algebra.
scientific article; zbMATH DE number 1375228
scientific article