On perfect Nikishin systems
The authors prove that an arbitrary Nikishin system of three measures is strongly perfect. It means that for any multiindices \((n_1,n_2,n_3)\), \(| n| =n_1+n_2+n_3,\) and for any Nikishin system of measures \(S=(s_1,s_2,s_3)\) (i.e., \(s_1=\sigma_1,s_2=\langle\sigma_1,\sigma_2\rangle,\) \(s_3= \langle\sigma_1,\langle\sigma_2,\sigma_3\rangle\rangle,\) \(\langle\sigma_1,\sigma_2\rangle(x):=\int\frac{d\sigma-2(t)}{x-t} d\sigma_1(x)\) and \(\sigma_1,\sigma_2,\sigma_3\) are finite Borel measures on \(\mathbb R\) with constant sign and mutually disjoint compact supports) there exists a polynomial \(Q_n\) of degree \(| n| \) such that \[ \int x^kQ_n(x)ds_j(x)=0,\quad k=0,1,\ldots,n_j-1;\quad j=1,2,3, \] and \(Q_n\) has exactly \(| n| \) simple zeros in the interior of Co(supp(\(s_1)).\) The result is applied to the convergence of the Hermite-Padé approximants relative to the system \(S.\)
scientific article; zbMATH DE number 2103476