Multipoint Padé-Type Approximants. Exact Rate of Convergence

Abstract

Abstract. We study the rate with which sequences of interpolating rational functions, whose poles are partially fi ed, approximate Markov-type analytic functions. Applications to interpolating quadratures are given.

Multipoint Rational Approximants with Preassigned Poles

“…This means thatq n is the orthonormal polynomial of degree n ¡ k(n) with respect to the measure jT n j 2 d¹; ¹ 2 Reg. They showed that the polynomialsp n =q n T n have the same n th root asymptotic behavior as p n de¯ned by (7), under fairly weak conditions on the sequence fT n g (under more restrictive conditions on ¹ this result was previously obtained in [2]).…”

“…hd¸. This condition was¯rst introduced in [2] and later used also in [1] and [5]. We also assume that…”

“…De¯nition 1. The Multipoint Pad ¶ e-type Approximant (MPTA) of1 with preassigned poles at the zeros of a given polynomial L 2 n , deg L n = k(n) • n, L n 6 0, which interpolates the function1 at the zeros of the polynomial A 2n with deg A 2n • 2n, given by (2), is the unique rational function…”

Quadrature and orthogonal rational functions

“…In [30] the authors could rely on known results about the asymptotic behavior of polynomials orthogonal with respect to fixed complex measures and their zeros to overcome these difficulties. For a general rational setting, a treatment similar to the one in Section 6 is given in [29,21,20]. The idea is as follows.…”

“…González-Vera et al prove in [29] the convergence of this type of quadrature formulas in the class of continuous functions satisfying a certain Lipschitz condition. Cala-Rodriguez and López-Lagomasino in [21] derive exact rates of convergence when approximating Markov-type analytic functions. In both of these papers, the intimate connection between multipoint Padé-type approximants and interpolatory quadrature formulas is explicitly exploited.…”