Padé-Type Approximants of Markov and Meromorphic Functions

Asymptotics for Stieltjes polynomials, padé-type approximants, and Gauss-Kronrod quadrature

“…We have considered the particular case w ≡ 1, α = 1, β = 2. Numerical experiments show that the zeros of the Stieltjes polynomials for this measure have an interesting behaviour; while the zeros of S 2n sit on [−2, −1] ∪ [1,2], those of S 2n+1 draw the level curve {ζ ∈ C : g Ω (ζ, ∞) = g Ω (0, ∞)}. Figure 1 shows the zeros of S n for n = 20, 21 (the small circles are the zeros of S 20 and the crosses the zeros of S 21 ).…”

“…Such results have direct application in the approximation of Markov functions by means of rational approximants with partially prescribed poles (Padé-type approximants in the terminology commonly used in recent years). Regarding such approximants, we refer to the papers [1]- [3] and the references therein. As a by-product of the results obtained in rational approximation, we give estimates of the rate of convergence of Gauss-Kronrod quadrature for functions which are analytic on a neighbourhood of the set of integration.…”

Convergence Rates of Padé and Padé-Type Approximants

“…On the other hand, if we use PTAs, for example for the case of functions analytic in C "2, 2=[&1, 1], we can use, in particular, any classical system of orthogonal polynomials on 2 (Legendre, Chebyshev, etc.) (see [8], p. 200, Theorem III b or [3], Theorem 3). We have these polynomials ready beforehand and so it takes no time or effort to calculate the denominator of the PTAs.…”

“…In particular, f may be a Cauchy type integral. Moreover, we have convergence for the whole class of functions analytic in the unbounded component of C "K, and even for the class of meromorphic functions in C "K (see [8], p. 200, Theorem III b for analytic functions and [3], Theorem 3 for the general case).…”

“…If we consider PTAs P n&1 ÂQ n of the same function (1) they give the rate (see [8], p. 200 or [3], Theorem 3)…”