Convergence of multipoint Padé-type approximants (Q5933479)

This is a very nice and interesting paper, extending a result by \textit{E. A. Rakhmanov} [Math. Sb., n. Ser. 33, 243-260 (1977; Zbl 0376.30011] on convergence of Padé approximants to NEWLINE\[NEWLINE f(z)=\int {d\mu(x)\over z-x}+r(z),\;z\in{\mathbb{C}}\setminus (S(\mu)\cup {\mathcal P}) NEWLINE\]NEWLINE (\(\mu\) a finite positive Borel measure with compact support \(S(\mu)\) on the real line, containing infinitely many points and \(r\) being a rational function with real coefficients whose poles (the set \(\mathcal P\)) are on \(\mathbb{C}\setminus I\;(I:\) the convex hull of \(S(\mu)\)) and \(r(\infty)=0\)) to the case of multi-point Padé approximation. NEWLINENEWLINENEWLINEIntroducing a sequence \(\{L_n\}_{n\geq 1}\) of monic polynomials with zeros in \(I\) and deg\( L_n=k(n)\leq n,\;n-k(n)>2d\) (\(d\): the degree of the denominator polynomial of \(r\)), the multi-point Padé-type approximants are the unique rational functions \(\Pi_n(f)=p_n/(q_nL_n^2)\) where \(\deg q_n\leq n-k(n)\), \(\deg p(n)\leq n+k(n)-1\), \(q_n\not\equiv 0\), \(q_nL_n^2f-p_n/w_n\) is holomorphic on \(\mathbb{C}\setminus (S(\mu)\cup {\mathcal P})\), \(q_nL_n^2f-p_n/w_n=O(1/z^{n-k(n)+1})\), \(z\rightarrow\infty\) (i.e. the zeros of \(L_n^2\) are the pre-assigned poles and \(\Pi_n\) interpolates \(f\) at the zeros of \(w_n\): a family of monic polynomials with \(\deg w_n=2n\) whose zeros lie in a compact set \(L\subset {\overline\mathbb{C}}\setminus (I\cup {\mathcal P})\) and are located symmetrically with respect to the real line).NEWLINENEWLINENEWLINEThe main result is then: If \(\{w_n\}\) has \(\nu\) as its asymptotic zero distribution (weak star limit of the normalized zero counting measure), \(\operatorname {cap}S(\mu)>0\) and either \(k(n)=o(n)\) or NEWLINE\[NEWLINE \limsup_{n\rightarrow\infty} \left\|\exp\biggl(-k(n)\int\log|z-t|d\nu(t)\biggr) L_n\right\|_{S(\mu)}^{1/k(n)} \leq\exp{(-F_w)} NEWLINE\]NEWLINE (\(F_w\) the modified Robin constant for \(w\)) then 1. for \(n\geq n_0\) the degree of \(q_n\) is \(n-k(n)\), the number of poles of \(\Pi_n(f)\) in \(\mathbb{C}\setminus I\) is then equal to the number of poles of \(r\) and the poles of \(\Pi_n(f)\) in \(\mathbb{C}\setminus I\) tend to those of \(r\) as \(n\rightarrow\infty\) and each pole of \(r\) attracts a number of poles equal to its order, 2. on compact subsets \(K\subset{\overline{\mathbb{C}}}\setminus (I\cup {\mathcal P})\) NEWLINE\[NEWLINE \limsup_{n\rightarrow\infty} \|f-\Pi_n(f)\|_K^{1/2n}\leq \|\exp\{-G_{\Omega}(\nu;\cdot)\}\|_K NEWLINE\]NEWLINE [\(G\): the Green potential of \(\nu\) on \(\Omega={\overline{\mathbb{C}}}\setminus S(\mu)\)].