On boundary angular derivatives of an analytic self-map of the unit disk
Let \(D\) be the unit disk. The Schur class \(\mathcal{S}\) consists of all analytic functions \(f:D\to \overline{D}\). I.~Schur gave a necessary and sufficient condition for the existence of \(f\in \mathcal{S}\) with given first Taylor coefficients \(s_0\), \(s_1,\ldots,s_{n-1}\). The author investigates the following problem \(BP_N\) (boundary analog of the Schur problem): For given \(s_0\), \(s_1,\ldots,s_{N}\), does there exists \(f\in \mathcal{S}\) satisfying \(f(z)=s_0+s_1(z-t_0)+\cdots+s_N(z-t_0)^N+o((z-t_0)^N)\) as \(z\to t_0\) non-tangentially? Here \(t_0\) is a fixed point in \(\partial D\). For \(n\leq N-1\) and \(1\leq j,l\leq n\), let \(\Psi_{jl}=0\) if \(l<j\), otherwise \[ \Psi_{jl}=(-1)^{l-1}\begin{pmatrix} l-1 \cr j-1\cr t_0^{l+j-1}\end{pmatrix}. \] For natural \(i,\, j\) such that \(i+j\leq N+1\), define \(p_{ij}=\sum_{r=1}^j\left(\sum_{l=1}^r s_{i+l-1}\Psi_{lr}\right)\overline{s}_{j-r}\). Let \(P_n=[p_{ij}]_{i,j=1}^n\), \(B_n=[p_{i,n+1}]_{1\leq i\leq n}\). The main result is Theorem 2.1. Let \(n\leq (N+1)/2\) be the greatest integer such that \(P_n\) is a positive semidefinite matrix. Then (1) The problem \(BP_N\) has a unique solution iff \(|s_0|=1\), \(P_n\) is singular, and either (a) \(n=(N+1)/2\) and \(\text{rank\,}P_n=\text{rank\,}P_{n-1}\), or (b) \(n=N/2\), \(p_{n+1,n}=\overline{p}_{n,n+1}\) and \(\text{rank\,}P_n=\text{rank\,}[P_nB_n]\). In this case the unique solution to the problem is a finite Blaschke product of degree equal to \(\text{rank\,}P_n\). (2) The problem \(BP_N\) has infinitely many solutions iff either (a) \(|s_0|<1\), or (b) \(|s_0|=1\), \(P_n>0\), and one of the following conditions holds: (i) \(n=(N+1)/2\), (ii) \(n=N/2\) and \(t_0(p_{n+1,n}-\overline{p}_{n,n+1})>0\). (3) Otherwise the problem has no solution.
scientific article; zbMATH DE number 5539120