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In the theory of Pad e approximation locally uniform convergence has been proved only for special classes of functions: for much larger classes convergence in capacity has been shown to hold true. The reason for one type of convergence to hold true, but the other one not, can be found in poles of the approximants that may occur apparently anywhere in the(More)
The asymptotic behavior of quadratic Hermite-Padé polynomials pn, qn, rn ∈ Pn of type I and pn, qn, rn ∈ P 2n of type II associated with the exponential function are studied. In the introduction the background of the definition of Hermite-Padé polynomials is reviewed. The quadratic Hermite-Padé polynomials pn, qn, rn ∈ Pn of type I are defined by the(More)
The convergence of (diagonal) sequences of rational interpolants to an analytic function is investigated. Problems connected with their definition are shortly discussed. Results about locally uniform convergence are reviewed. Then the convergence in capacity is studied in more detail. Here, a central place is taken by a theorem about the convergence in(More)
Let α > 0 and ψ (x) = x α. Let Sn,α be a polynomial of degree n determined by the biorthogonality conditions Z 1 0 We explicitly determine Sn,α and discuss some other properties, including their zero distribution. We also discuss their relation to the Sidi polynomials. §1. Introduction and Results Let ψ : (0, 1) → R be a strictly increasing continuous(More)
Let f (z)= (t&z) &1 d+(t) be a Markov function, where + is a positive measure with compact support in R. We assume that supp(+)/(&1, 1), and investigate the best rational approximants to f in the Hardy space H 0 2 (V), where V :=[z # C | |z|>1] and H 0 2 (V) is the subset of functions f # H 2 (V) with f ()=0. The central topic of the paper is to obtain(More)
We obtain strong and uniform asymptotics in every domain of the complex plane for the scaled polynomials a(3nz), b(3nz), and c(3nz) where a, b, and c are the type II Hermite-Padé approximants to the exponential function of respective degrees 2n + 2, 2n and 2n, defined by a(z)e −z − b(z) = O(z 3n+2) and a(z)e z − c(z) = O(z 3n+2) as z → 0. Our analysis(More)
The Baker–Gammel-Wills Conjecture states that if a function f is mero-morphic in a unit disk D, then there should, at least, exist an infinite subsequence N ⊆ N such that the subsequence of diagonal Padé approximants to f developed at the origin with degrees contained in N converges to f locally uniformly in D/{poles of f }. Despite the fact that this(More)
Let f be holomorphically continuable over the complex plane except for finitely many branch points contained in the unit disk. We prove that best rational approximants to f of degree n, in the L2-sense on the unit circle, have poles that asymptotically distribute according to the equilibrium measure on the compact set outside of which f is single-valued and(More)