Boundary asymptotic expansions of analytic self-mappings of the unit disk (Q2807104)

Let \(\mathcal S=\mathrm{Hol}(\mathbb D,\overline{\mathbb D})\) consist of the self-mappings of the unit disk \(\mathbb D\) plus the constant unimodular functions. This paper addresses the following general problem (named \(\mathrm{BIP}(\zeta, k,s)\) henceforth). Let \(\zeta=(\zeta_1,\dots,\zeta_N)\) be an \(N\)-tuple of distinct boundary points \(\zeta_i\in\partial\mathbb D\), along with a \(k\)-tuple of multiplicities \(k=(k_1,\dots,k_N)\), \(k_i\in\mathbb N\), and a doubly indexed collection of values \(s=\{s_{i,j}\}_{i=1,\dots,N}^{j=0,\dots,k_i}\). Does there exist \(F\in\mathcal S\) having asymptotic expansions NEWLINE\[NEWLINE F(z)=\sum_{j=0}^{k_i} s_{i,j} (z-\zeta_i)^j + o(|z-\zeta_i|^{k_i+1}),\;\;i=1,\dots,N, NEWLINE\]NEWLINE as \(z\rightarrow\zeta_i\) non-tangentially?NEWLINENEWLINEThese asymptotic identities are equivalent to the existence of the non-tangentially boundary limits NEWLINE\[NEWLINE F_j(\zeta_i)=\angle \lim_{z\to\zeta_i} \frac{F^{(j)}(z)}{j!}=s_{i,j},\;\;\;\;i=1,\dots,N,\;j=0,\dots,k_i. NEWLINE\]NEWLINENEWLINENEWLINEThe paper extends the results given previously by the first author for \(N=1\) [J. Approx. Theory 163, No. 4, 568--589 (2011; Zbl 1256.30019)] about the solvability and the determinacy (whether there exists one or more solutions) of the above system. The results, which are very precise, are given essentially in terms of the positive-definiteness of the so-called Pick matrix of the \(\mathrm{BIP}(\zeta,k,s)\) problem, which is constructed with the problem data \(\zeta\), \(k\), \(s\) and is too technical to be detailed here.