Nuttall's integral equation and Bernshtein's asymptotic formula for a complex weight (Q2832319)

The main results of the paper are:NEWLINENEWLINE{Theorem 1.} Suppose that the functions \(F_n\), \(n\in{\mathbb N}\) are not identically equal to zero, holomorphic in the disc \(|z|<1\) and continuous in the closed disc \(|z|\leq 1\). Furthermore, assume NEWLINE\[NEWLINEF_n(z)=\int_{|t|=1}\,{F_n(1/t)g(t)\over t^{2n}(t-z)}dt+C_n,\quad |z|<1,NEWLINE\]NEWLINE for a given function \(g\) satisfying the Dini-Lipschitz condition on \(|t|=1\) with constants \(C>0\) and \(\lambda>1\), \(C_n\) a constant. Then NEWLINE\[NEWLINEF_n(z)=C_n(1+o(1)),\quad n\rightarrow\infty,NEWLINE\]NEWLINE uniformly in \(|z|\leq 1\), where \(o(1)={\mathcal O}((\log{n})^{-\mu})\), \(\mu=\lambda-1\). NEWLINENEWLINENEWLINENEWLINE The Dini-Lipschitz condition with constants \(C>0\) and \(\lambda>1\) on the unit circle \(\Gamma\) can be formulated as NEWLINE\[NEWLINE|g(t)-g(t')|\leq C\left(\log{1\over |t-t'|}\right)^{-\lambda},\quad \lambda>1,\quad t,t'\in \Gamma.NEWLINE\]NEWLINENEWLINENEWLINE{Theorem 2.} Suppose that the weight \(\sigma(x)\) satisfies the Dini-Lipschitz condition on \(\Delta:=[-1,1]\) with constants \(C>0\) and \(\lambda >1\). Then the Padé polynomials \(P_{n,j}(z;f),\;j=1,2\) for NEWLINE\[NEWLINEf(z)={1\over 2\pi i}\,\int_{-1}^1\,{\sigma(x)\over w^{+}(x)}{dx\over x-z},\quad z\not\in \DeltaNEWLINE\]NEWLINE are uniquely determined (up to normalization) for all sufficiently large \(n\). After an appropriate normalization and for arbitrary \(\delta >0\) we have the following uniform asymptotic formulae NEWLINE\[NEWLINEP_{n,j}(t)=t^n\Pi_j(t)(1+o(1)),\quad |t|>1+\delta,\quad n\rightarrow\infty,NEWLINE\]NEWLINE NEWLINE\[NEWLINEP_{n,j}(t)=t^n\Pi_j(t)+t^{-n}\Pi_j(t^{-1})(1+o(1)),\quad |t|=1,\quad n\rightarrow\infty,NEWLINE\]NEWLINE where \(o(1)={O}\big((\log{n})^{1-\lambda}\big) \). NEWLINENEWLINENEWLINE Here \(w(z)=(z^2-1)^{1/2}\), where the branch of the multi-valued root-function for \(z\in D:={\mathbb C}\setminus [-1,1]\) is chosen in such a way that NEWLINE\[NEWLINEw(z)=(z^2-1)^{1/2}\sim z,\quad z\rightarrow\infty,NEWLINE\]NEWLINE and \(w^{+}(x)\) is defined as usual by NEWLINE\[NEWLINEw^{+}(x):=w(x+i\cdot 0):=\lim_{\varepsilon\downarrow 0}\,w(x+i\varepsilon),\quad x\in (-1,1).NEWLINE\]NEWLINENEWLINENEWLINEFor the definitions of \(\Pi_1\) and \(\Pi_2\) one should consult the paper.