A de Montessus type convergence study of a least-squares vector-valued rational interpolation procedure (original) (raw)

A de Montessus type convergence study for a vector-valued rational interpolation procedure

Israel Journal of Mathematics, 2008

In a recent paper of the author [8], three new interpolation procedures for vector-valued functions F (z), where F : C → C N , were proposed, and some of their algebraic properties were studied. In the present work, we concentrate on one of these procedures, denoted IMMPE, and study its convergence properties when it is applied to meromorphic functions. We prove de Montessus and Koenig type theorems in the presence of simple poles when the points of interpolation are chosen appropriately. We also provide simple closed-form expressions for the error in case the function F (z) in question is itself a vector-valued rational function whose denominator polynomial has degree greater than that of the interpolant.

A new approach to vector-valued rational interpolation

Journal of Approximation Theory, 2004

In this work we propose three different procedures for vector-valued rational interpolation of a function F(z), where F : C → C N , and develop algorithms for constructing the resulting rational functions. We show that these procedures also cover the general case in which some or all points of interpolation coalesce. In particular, we show that, when all the points of interpolation collapse to the same point, the procedures reduce to those presented and analyzed in an earlier paper [J. Approx. Theory 77 (1994) 89] by the author, for vector-valued rational approximations from Maclaurin series of F(z). Determinant representations for the relevant interpolants are also derived.

Rational interpolation: I. Least square convergence

Given a positive bounded Borel measure µ on the interval [−1, 1], we provide convergence results in L µ 2 -norm to a function f of its sequence of interpolating rational functions at the nodes of rational Gauss-type quadrature formulas associated with the measure µ. For this, we use the connection between rational Gauss-type quadrature formulas on [−1, 1] and rational Szegő quadrature formulas associated with a positive symmetric Borel measureμ on the complex unit circle.

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.

The linear pencil approach to rational interpolation

Journal of Approximation Theory, 2010

It is possible to generalize the fruitful interaction between (real or complex) Jacobi matrices, orthogonal polynomials and Padé approximants at infinity by considering rational interpolants, (bi-)orthogonal rational functions and linear pencils zB − A of two tridiagonal matrices A, B, following Spiridonov and Zhedanov.

A new realization of rational functions, with applications to linear combination interpolation

2015

We introduce the following linear combination interpolation problem (LCI): Given N distinct numbers w_1,... w_N and N+1 complex numbers a_1,..., a_N and c, find all functions f(z) analytic in a simply connected set (depending on f) containing the points w_1,...,w_N such that ∑_u=1^Na_uf(w_u)=c. To this end we prove a representation theorem for such functions f in terms of an associated polynomial p(z). We first introduce the following two operations, (i) substitution of p, and (ii) multiplication by monomials z^j, 0< j < N. Then let M be the module generated by these two operations, acting on functions analytic near 0. We prove that every function f, analytic in a neighborhood of the roots of p, is in M. In fact, this representation of f is unique. To solve the above interpolation problem, we employ an adapted systems theoretic realization, as well as an associated representation of the Cuntz relations (from multi-variable operator theory.) We study these operations in reprodu...

The interpolation formula for a class of meromorphic functions

Journal of Approximation Theory, 2013

In this paper we consider a class of functions f (z) (z ∈ C) meromorphic in the half-plane Re z 1 2 , holomorphic in 0 < Re z < 1 2 , continuous on Re z = 0, and satisfying a suitable Carlson-type asymptotic growth condition. First we prove that the position and the residue of the poles of f (z) can be obtained from the samples of f (z) taken at the positive half-integers. In particular, the positions of the poles are shown to be the roots of an algebraic equation. Then we give an interpolation formula for f (x + 1 2 ) (x = Re z) that incorporates the information on the poles (i.e., position and residue) and which is proved to converge to the true function uniformly on x x 0 > − 1 2 as the number of samples tends to infinity and the error on the samples goes to zero. An illustrative numerical example of interpolation of a Runge-type function is also given.

An interpolation algorithm for orthogonal rational functions

Journal of Computational and Applied Mathematics, 2004

Let A = {α 1 , α 2 , . . . } be a sequence of numbers on the extended real lineR = R ∪ {∞} and µ a positive bounded Borel measure with support in (a subset of)R. We introduce rational functions φ n with poles {α 1 , . . . , α n } that are orthogonal with respect to µ (if all poles are at infinity, we recover the polynomial situation). It is well known that under certain conditions on the location of the poles, the system {φ n } is regular such that the orthogonal functions satisfy a three term recurrence relation similar to the one for orthogonal polynomials.

On a Fast Convergence of the Rational-Trigonometric-Polynomial Interpolation

We consider the convergence acceleration of the Krylov-Lanczos interpolation by rational correction functions and investigate convergence of the resultant parametric rational-trigonometric-polynomial interpolation. Exact constants of asymptotic errors are obtained in the regions away from discontinuities, and fast convergence of the rational-trigonometric-polynomial interpolation compared to the Krylov-Lanczos interpolation is observed. Results of numerical experiments confirm theoretical estimates and show how the parameters of the interpolations can be determined in practice.