Convergence of Rational Interpolants with Preassigned Poles

Self Cite

We investigate the following problem: For which open simply connected domains do there exist interpolation schemes (a set of interpolation points) such that for any analytic function defined in the domain the corresponding interpolating polynomials converge to the function when the degree of the polynomials tends to infinity? We also study similar problems for rational interpolants. These problems are connected to the balayage (sweeping out) problems of measures. Academic Press

Section: Results On Balayagementioning

confidence: 99%

Balayage and Convergence of Rational Interpolants

Ambroladze

Wallin

1999

Self Cite

“…This type of problem has been studied by Walsh (see [8]) and Bagby (see [4]) and more recently by Ambroladze and Wallin in the papers [1], [2] and [3]. If we consider interpolating polynomials (which corresponds to the case where all poles are at infinity) on a bounded simply connected domain D … C, and apply results from [1] and [3] we have the following.…”

Section: Introductionmentioning

confidence: 95%

“…If we consider interpolating polynomials (which corresponds to the case where all poles are at infinity) on a bounded simply connected domain D … C, and apply results from [1] and [3] we have the following.…”

Section: Introductionmentioning

confidence: 98%

Rational and Polynomial Interpolation of Analytic Functions with Restricted Growth

Gustafsson

2001

Let f be an analytic function on a domain D … C 2 {.} and r n the rational function of degree n with poles at the pointsA fundamental question is whether it is possible to choose the points A n and B n so that r n converges locally uniformly to f on D for every analytic function f on D. In some situations the interpolation points must be allowed to approach the boundary of D as n tends to infinity and then we cannot obtain convergence for every analytic f on D. If we restrict the growth of f(z) when z goes to the boundary of D, we still have some positive convergence results that we prove here.

Convergence of Multipoint Padé-type Approximants

Ysern

Lagomasino

2001

Let + be a finite positive Borel measure whose support is a compact subset K of the real line and let I be the convex hull of K. Let r denote a rational function with real coefficients whose poles lie in C"I and r( )=0. We consider multipoint rational interpolants of the functionwhere some poles are fixed and others are left free. We show that if the interpolation points and the fixed poles are chosen conveniently then the sequence of multipoint rational approximants converges geometrically to f in the chordal metric on compact subsets of C "I. Academic Press