Perturbation of nodes and poles in certain rational interpolants (original) (raw)

On uniform convergence of rational, Newton-Pad� interpolants of type (n, n) with free poles asn??

Numerische Mathematik, 1989

Let f be meromorphic in 112. We show that there exists a sequence of distinct interpolation points {zj}~% ~, and for n > 1, rational functions R,(z) of type (n, n) solving the Newton-Pad6 (Hermite) interpolation problem, R,(zi)=f(zj), j=l,2 .... 2n+l, and such that for each compact subset K of (I2 omitting poles of f, we have lim IIf-R.Q~tK~=O. n~o0 Extensions are presented to the case where f(z) is meromorphic in a given open set with certain additional properties, and related results are discussed.

On the Lebesgue constant of Berrut’s rational interpolant at equidistant nodes

Journal of Computational and Applied Mathematics, 2011

It is well known that polynomial interpolation at equidistant nodes can give bad approximation results and that rational interpolation is a promising alternative in this setting. In this paper we confirm this observation by proving that the Lebesgue constant of Berrut's rational interpolant grows only logarithmically in the number of interpolation nodes. Moreover, the numerical results show that the Lebesgue constant behaves similarly for interpolation at Chebyshev as well as logarithmically distributed nodes.

On Lagrange interpolation at disturbed roots of unity

Transactions of the American Mathematical Society, 1993

Let znk = eu"k, 0 < tn0 < ■ ■ ■ < t"" < 2n, f a function in the disc algebra A , and ßn = max{|f"fc-2kn/{n + 1)|: 0 < k < n}. Denote by L"{f; •) the polynomial of degree n that agrees with / at {znk: k = 0, ... , n}. In this paper, we prove that for every p, 0 < p < oo , there exists a Sp > 0, such that \\Ln{f; •)-f\\p = 0(o)(f; ¿)) whenever p" < Sp/{n + 1). It must be emphasized that Sp necessarily depends on p, in the sense that there exists a family {znk: k = 0, ... , «} with ß" = ô2/(n + 1) and such that \\L"{f; •)-f\\2 = 0{ca{f; 1)) for all / € A , but sup{||L"(/;-) 11^7 : / G A , \\f\\oc = 1} diverges for sufficiently large values of p. In establishing our estimates, we also derive a Marcinkiewicz-Zygmund type inequality for {znk} .

Equisummability of certain sequences of Hadamard products of Taylor sections and interpolatory polynomials

Methods and Applications of Analysis, 1994

In [1, 2] the classical equiconvergence theorem of Walsh was extended by the application of summability methods in order to enlarge the disk of equiconvergence to regions of equisummability. A further generalization was achieved in [3], where sequences of Hadamard products of a fixed power series with interpolatory polynomials were considered. The aim of this paper is to continue this work by investigating commutators of interpolatory polynomials and Hermite interpolatory polynomials.

On the Convergence of (0,1,2) Interpolation

2010

For the Hermite interpolation polynomial, Hm(x) we prove for any function f ∈ C(2q)([−1, 1]) and any s = 0, 1, 2, . . . , q, where q is a fixed integer that |H m (x) − f (x)| = O(1)ω( 1 m , f ) log n n2q−2s . Here m is defined by m = 3n− 1. If f ∈ C(q)([−1, 1]), then |H m − f (x)| = O(1)ω( 1 m , f ) log n (1 − x2)q/2 for x ∈ (−1, 1). 2000 Mathematics Subject Classification: 41A05.

On P ´ Al-Type Interpolation II

In this paper, we study the convergence of Pál-type interpolation on two sets of non-uniformly distributed zeros on the unit circle, which are obtained by projecting vertically the nodes of the real line.