Herbert Stahl

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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)
where ‖ · ‖K denotes the sup norm on K ⊆ R. It is well known that the best approximant r∗ mn exists and is unique within Rmn (cf. [Me, §§9.1, 9.2] or [Ri, §5.1]). The unique existence also holds in the special case (n = 0) of best polynomial approximants. Since fα(x) := |x|α is an even function on [−1, 1], the same is true for its unique approximant r∗ mn =(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)
The asymptotic behavior of quadratic Hermite-Padé polynomials pn, qn, rn ∈ Pn of type I and pn, qn, rn ∈ P2n 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)
Let α > 0 and ψ (x) = x. Let Sn,α be a polynomial of degree n determined by the biorthogonality conditions Z 1 0 Sn,αψ j = 0, j = 0, 1, . . . , n− 1. We explicitly determine Sn,α and discuss some other properties, including their zero distribution. We also discuss their relation to the Sidi polynomials. §
Let f (z)= (t&z) 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 2(V), where V :=[z # C | |z|>1] and H 2(V) is the subset of functions f # H2(V) with f ( )=0. The central topic of the paper is to obtain asymptotic(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)