Padé approximants for functions with branch points -- strong asymptotics of Nuttall-Stahl polynomials (Q265873)

This paper describes the asymptotics of diagonal Padé approxiamants of algebraic functions of a special type.NEWLINENEWLINEThe underlying function \(f\) is holomorphic at infinity and extends analytically (in a multivalued way) along any path in the extended complex plane that omits a finite number of points \(A=\{a_k\}\) (at least two), subject to two general restrictions.NEWLINENEWLINE\textbf{1.} The character of the branch points is assumed to be \textit{algebro-logarithmic}, i.e., in a small neighborhood of the \(a_k\) the function has a representation NEWLINE\[NEWLINEf(z)=h_1(z)\psi(z)+h_2(z),\;\psi(z)=\begin{cases} (z-a_k)^{\alpha(a_k)}\\ \log{(z-a_k)}\end{cases}, NEWLINE\]NEWLINE where \(-1<\alpha(a_k)<0\) and \(h_1,h_2\) are holomorphic around \(a_k\).NEWLINENEWLINE\textbf{2.} The \textit{extremal domain} \(d^{\ast}\) for \(f\) in the sense of \textit{H. Stahl} [Complex Variables, Theory Appl. 4, 311--324 (1985; Zbl 0542.30027), ibid. 4, 339--354 (1985; Zbl 0542.30029)] satisfies NEWLINE\[NEWLINE\mathbb{C}\setminus D^{\ast}=E\cup \bigcup_{k}\Delta_k,NEWLINE\]NEWLINE where \(\bigcup_k \Delta_k\) is a finite union of open analytic Jordan arcs and \(E\) is a finite set of points such that each element of \(E\) is an endpoint for at least one arc \(\Delta_k\). The condition referred to as GP (\textit{generic position}) is nowNEWLINENEWLINE(i)\ \ each point in \(E\cup A\) is incident with exactly one arc from \(\bigcup_{k}\Delta_k\),NEWLINENEWLINE(ii) each point in \(E\setminus A\) ia incident with exactly three points in \(\bigcup_{k}\Delta_k\).NEWLINENEWLINEThe main result is then stated in Theorem 2.11; explicitly reproducing this theorem is outside the scope of a review: it would take several pages to formulate the concepts needed.NEWLINENEWLINEFor the initiated: the work is an extension of the proof of \textit{J. Nuttall}'s conjecture (in [in: Pade and ration. Approx., Theory and Appl., Proc. int. Symp. Tampa/Florida 1976, 101--109 (1977; Zbl 0368.41012)]) on convergence in logarithmic capacity away from the cuts characterized by minimal logarithmic capacity.NEWLINENEWLINEThe layout of the paper is as follows:NEWLINENEWLINE\S1 Introduction (6 pages)NEWLINENEWLINE\S2 Main results (12 pages)NEWLINENEWLINE\S3 Extremal domains (9 pages)NEWLINENEWLINE\S4 Riemann surface (5 pages)NEWLINENEWLINE\S5 Boundary problems on \(L\) (5 pages)NEWLINENEWLINE\S6 Szegő functions (5 pages)NEWLINENEWLINE\S7 Riemann-Hilbert problem (6 pages)NEWLINENEWLINE\S8 Asymptotic analysis (14 pages)NEWLINENEWLINEReferences (3 pages; 58 items)