scholarly journals Rational Interpolation: Jacobi’s Approach Reminiscence

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1401
Author(s):  
Alexei Uteshev ◽  
Ivan Baravy ◽  
Elizaveta Kalinina
Keyword(s):  
Interpolation Problem ◽  
Symmetric Functions ◽  
Rational Functions ◽  
Rational Interpolation ◽  
Effective Procedure ◽  
Recursive Procedure ◽  
Data Set ◽  
Rational Interpolants ◽  
Extra Point

We treat the interpolation problem {f(xj)=yj}j=1N for polynomial and rational functions. Developing the approach originated by C. Jacobi, we represent the interpolants by virtue of the Hankel polynomials generated by the sequences of special symmetric functions of the data set like {∑j=1Nxjkyj/W′(xj)}k∈N and {∑j=1Nxjk/(yjW′(xj))}k∈N; here, W(x)=∏j=1N(x−xj). We also review the results by Jacobi, Joachimsthal, Kronecker and Frobenius on the recursive procedure for computation of the sequence of Hankel polynomials. The problem of evaluation of the resultant of polynomials p(x) and q(x) given a set of values {p(xj)/q(xj)}j=1N is also tackled within the framework of this approach. An effective procedure is suggested for recomputation of rational interpolants in case of extension of the data set by an extra point.

scholarly journals Rational interpolation of the function |x|α by an extended system of Chebyshev – Markov nodes

2020 ◽  
Vol 55 (4) ◽  
pp. 391-405
Author(s):  
E. A. Rovba ◽  
V. Yu. Medvedeva
Keyword(s):  
Rational Functions ◽  
Rational Interpolation ◽  
Asymptotic Equality ◽  
Fixed Number ◽  
Theoretical Research ◽  
Extended System ◽  
Uniform Approximations ◽  
Upper Estimate ◽  
Interpolation Functions ◽  
Interpolation Nodes

In this paper, we study the approximations of a function |x|α, α > 0 by interpolation rational Lagrange functions on a segment [–1,1]. The zeros of the even Chebyshev – Markov rational functions and a point x = 0 are chosen as the interpolation nodes. An integral representation of an interpolation remainder and an upper bound for the considered uniform approximations are obtained. Based on them, a detailed study is made:a) the polynomial case. Here, the authors come to the famous asymptotic equality of M. N. Hanzburg;b) at a fixed number of geometrically different poles, the upper estimate is obtained for the corresponding uniform approximations, which improves the well-known result of K. N. Lungu;c) when approximating by general Lagrange rational interpolation functions, the estimate of uniform approximations is found and it is shown that at the ends of the segment [–1,1] it can be improved.The results can be applied in theoretical research and numerical methods. 

scholarly journals Solution of a multiple Nevanlinna–Pick problem for Schur functions via orthogonal rational functions

Analysis ◽  
2008 ◽  
Vol 28 (1) ◽  
Author(s):  
Adhemar Bultheel ◽  
Andreas Lasarow
Keyword(s):  
Unique Solution ◽  
Interpolation Problem ◽  
Schur Functions ◽  
Rational Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Orthogonal Rational Functions ◽  
All Solutions ◽  
Pick Problem ◽  
Complex Valued

An interpolation problem of Nevanlinna–Pick type for complex-valued Schur functions in the open unit disk is considered. We prescribe the values of the function and its derivatives up to a certain order at finitely many points. Primarily, we study the case that there exist many Schur functions fulfilling the required conditions. For this situation, an application of the theory of orthogonal rational functions is used to characterize the set of all solutions of the problem in question. Moreover, we treat briefly the case of exactly one solution and present an explicit description of the unique solution in that case.

scholarly journals SCHUR–NEVANLINNA SEQUENCES OF RATIONAL FUNCTIONS

2007 ◽  
Vol 50 (3) ◽  
pp. 571-596 ◽  
Author(s):  
Adhemar Bultheel ◽  
Andreas Lasarow
Keyword(s):  
Unit Circle ◽  
Interpolation Problem ◽  
Schur Functions ◽  
Rational Functions ◽  
The Other ◽  
Complex Numbers ◽  
Orthogonal Rational Functions ◽  
All Solutions ◽  
The One

AbstractWe study certain sequences of rational functions with poles outside the unit circle. Such kinds of sequences are recursively constructed based on sequences of complex numbers with norm less than one. In fact, such sequences are closely related to the Schur–Nevanlinna algorithm for Schur functions on the one hand, and to orthogonal rational functions on the unit circle on the other. We shall see that rational functions belonging to a Schur–Nevanlinna sequence can be used to parametrize the set of all solutions of an interpolation problem of Nevanlinna–Pick type for Schur functions.

scholarly journals On a Bivariate Generalization of Berrut’s Barycentric Rational Interpolation to a Triangle

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2481
Author(s):  
Len Bos ◽  
Stefano De Marchi
Keyword(s):  
Rational Interpolation ◽  
Rational Interpolants ◽  
Barycentric Rational Interpolation

We discuss a generalization of Berrut’s first and second rational interpolants to the case of equally spaced points on a triangle in R2.

scholarly journals Minimal solutions of the rational interpolation problem

2020 ◽  
pp. 413-429
Author(s):  
Teresa Cortadellas Benítez ◽  
Carlos D'Andrea ◽  
Eulàlia Montoro
Keyword(s):  
Interpolation Problem ◽  
Rational Interpolation

scholarly journals Cumulants of Jack symmetric functions and b-conjecture (extended abstract)

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Maciej Dolega ◽  
Valentin Féray
Keyword(s):  
Nonnegative Integer ◽  
Symmetric Functions ◽  
Rational Functions ◽  
Independent Interest ◽  
Jack Polynomials ◽  
Factorization Property ◽  
Integer Coefficients ◽  
Continuous Deformation ◽  
Generating Series ◽  
International Audience

International audience Goulden and Jackson (1996) introduced, using Jack symmetric functions, some multivariate generating series ψ(x, y, z; t, 1 + β) that might be interpreted as a continuous deformation of the rooted hypermap generating series. They made the following conjecture: coefficients of ψ(x, y, z; t, 1+β) are polynomials in β with nonnegative integer coefficients. We prove partially this conjecture, nowadays called b-conjecture, by showing that coefficients of ψ(x, y, z; t, 1 + β) are polynomials in β with rational coefficients. Until now, it was only known that they are rational functions of β. A key step of the proof is a strong factorization property of Jack polynomials when α → 0 that may be of independent interest.

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