Interpolation properties of Hermite–Padé polynomials

Abstract

where σ1 is a positive measure with support supp σ1 on a compact set E ⊂ R and h ∈ H (E) is a holomorphic function on E. If h(z) = σ̂2(z), where σ2 is a positive measure with support supp σ2 ⊂ F , where F ⊂ R \ E is a compact set, then the pair of functions f1, f2 forms a Nikishin system (see [6], and also [7], [5], [10], and the bibliography therein). Let Qn,j , j = 0, 1, 2, be the Hermite–Padé polynomials of the first type for the collection [1, f1, f2] with multi-index n = (n − 1, n, n), which means that deg Qn,j ⩽ n and (Qn,0 + Qn,1f1 + Qn,2f2)(z) = O(z−2n−2), z →∞. (2) For an arbitrary polynomial Q ∈ C[z] \ 0, let