Lagrange interpolation with constraints

Uniform convergence of Lagrange interpolation at the zeros of Jacobi polynomials in the presence of constraints is investigated. We show that by a simple procedure it is always possible to transform the matrices of these zeros into matrices such that the corresponding Lagrange interpolating polynomial respecting the given constraints well approximates a given function and its derivatives.     

Intertwining unisolvent arrays for multivariate Lagrange interpolation

Generalizing a classical idea of Biermann, we study a way of constructing a unisolvent array for Lagrange interpolation in C^n+m out of two suitably ordered unisolvent arrays respectively in Cⁿ and C^m. For this new array, important objects of Lagrange interpolation theory (fundamental Lagrange polynomials, Newton polynomials, divided difference operator, vandermondian, etc.) are computed. AMS subject classification 41A05, 41A63     

Extended Lagrange interpolation in weighted uniform norm

The author studies the uniform convergence of extended Lagrange interpolation processes based on the zeros of Generalized Laguerre polynomials.     

The L p-weighted Lagrange interpolation on Markov--Sonin zeros

Summary We study a truncated interpolation process based on the zeros of the Markov--Sonin polynomials and give convergence results in some subspaces of the L^p weighted spaces.     

On polynomial interpolation of two variables

Polynomial interpolation of two variables based on points that are located on multiple circles is studied. First, the poisedness of a Birkhoff interpolation on points that are located on several concentric circles is established. Second, using a factorization method, the poisedness of a Hermite interpolation based on points located on various circles, not necessarily concentric, is established. Even in the case of Lagrange interpolation, this gives many new sets of poised interpolation points.     

Optimal recovery of interpolation operators in Hardy spaces

We continue studies begun by C.A. Micchelli and T.J. Rivlin on optimal recovery in H^p spaces. The feature operators are various interpolation operators drawn from the theory of Walsh equiconvergence, as are the information sets. The theory is of interest in that it identifies linear algorithms which might not otherwise be isolated for study or used as approximations of the feature operators. In some cases, we can identify the optimal algorithm although we cannot explicitly determine the exact order of the approximation that it achieves. For Charles Micchelli on his sixtieth birthday, with appreciation Mathematics subject classifications (2000) 41A05, 30B30. A. Sharma: Deceased.     

Approximating exponential and logarithmic functions using polynomial interpolation

This article takes a closer look at the problem of approximating the exponential and logarithmic functions using polynomials. Either as an alternative to or a precursor to Taylor polynomial approximations at the precalculus level, interpolating polynomials are considered. A measure of error is given and the behaviour of the error function is analysed. The results of interpolating polynomials are compared with those of Taylor polynomials.     

On the construction of interpolation mesh surfaces

A method for constructing two-dimensional interpolation mesh functions is proposed that is more flexible than the classical cubic spline method because it makes it possible to construct interpolation surfaces that fit the given function at specified points by varying certain parameters. The method is relatively simple and is well suited for practical implementation.     

Discrete Sturm–Liouville problems, Jacobi matrices and Lagrange interpolation series

The close relationship between discrete Sturm–Liouville problems belonging to the so-called limit-circle case, the indeterminate Hamburger moment problem and the search of self-adjoint extensions of the associated semi-infinite Jacobi matrix is well known. In this paper, all these important topics are also related with associated sampling expansions involving analytic Lagrange-type interpolation series.     

Convex Optimal Designs for Compound Polynomial Extrapolation

The extrapolation design problem for polynomial regression model on the design space [–1,1] is considered when the degree of the underlying polynomial model is with uncertainty. We investigate compound optimal extrapolation designs with two specific polynomial models, that is those with degrees |m, 2m}. We prove that to extrapolate at a point z, |z| > 1, the optimal convex combination of the two optimal extrapolation designs | _m ^* (z), _2m ^* (z)} for each model separately is a compound optimal extrapolation design to extrapolate at z. The results are applied to find the compound optimal discriminating designs for the two polynomial models with degree |m, 2m}, i.e., discriminating models by estimating the highest coefficient in each model. Finally, the relations between the compound optimal extrapolation design problem and certain nonlinear extremal problems for polynomials are worked out. It is shown that the solution of the compound optimal extrapolation design problem can be obtained by maximizing a (weighted) sum of two squared polynomials with degree m and 2m evaluated at the point z, |z| > 1, subject to the restriction that the sup-norm of the sum of squared polynomials is bounded.     

Chebyshev rational interpolation

The Lanczos method and its variants can be used to solve efficiently the rational interpolation problem. In this paper we present a suitable fast modification of a general look-ahead version of the Lanczos process in order to deal with polynomials expressed in the Chebyshev orthogonal basis. The proposed approach is particularly suited for approximating analytic functions by means of rational interpolation at certain nodes located on the boundary of an elliptical region of the complex plane. In fact, in this case it overcomes some of the numerical difficulties which limited the applicability of the look-ahead Lanczos process for determining the coefficients both of the numerators and of the denominators with respect to the standard power basis. This revised version was published online in August 2006 with corrections to the Cover Date.     

Polynomial interpolation over quaternions

Interpolation theory for complex polynomials is well understood. In the non-commutative quaternionic setting, the polynomials can be evaluated “on the left” and “on the right”. If the interpolation problem involves interpolation conditions of the same (left or right) type, the results are very much similar to the complex case: a consistent problem has a unique solution of a low degree (less than the number of interpolation conditions imposed), and the solution set of the homogeneous problem is an ideal in the ring H[z]$\mathbb{H}[z]$. The problem containing both “left” and “right” interpolation conditions is quite different: there may exist infinitely many low-degree solutions and the solution set of the homogeneous problem is a quasi-ideal in H[z]$\mathbb{H}[z]$.     

整系数多项式有理根一个新求法的再探讨

设f (x)为整系数多项式,α为有理数,对n个不同的整数t1,…,tn,gα(tk) =f (tk)tk-α都是整数,那么α是f (x)的根的充要条件是f (t) =∑ni=1∏1≤j≤nj≠it-tjti-tjgα(ti) ( t∈Z) .     

Local lagrange interpolation with cubic splines on tetrahedral partitions

We describe an algorithm for constructing a Lagrange interpolation pair based on C¹ cubic splines defined on tetrahedral partitions. In particular, given a set of points , we construct a set P containing and a spline space based on a tetrahedral partition whose set of vertices include such that interpolation at the points of P is well-defined and unique. Earlier results are extended in two ways: (1) here we allow arbitrary sets , and (2) the method provides optimal approximation order of smooth functions.     

Polynomial interpolation on the unit sphere II

The problem of interpolation at (n+1)² points on the unit sphere by spherical polynomials of degree at most n is proved to have a unique solution for several sets of points. The points are located on a number of circles on the sphere with even number of points on each circle. The proof is based on a method of factorization of polynomials. Dedicated to Mariano Gasca on the occasion of his 60th birthday The second author was supported by the Graduate Program Applied Algorithmic Mathematics of the Munich University of Technology. The work of the third author was supported in part by the National Science Foundation under Grant DMS-0201669.     

On some aspects of multivariate polynomial interpolation

The purpose of this paper is to present some aspects of multivariate Hermite polynomial interpolation. We do not focus on algebraic considerations, combinatoric and geometric aspects, but on explicitation of formulas for uniform and non-uniform bivariate interpolation and some higher dimensional problems. The concepts of similar and equivalent interpolation schemes are introduced and some differential aspects related to them are also investigated. This revised version was published online in June 2006 with corrections to the Cover Date.     

Simultaneous approximation by hermite interpolation of higher order

The authors consider a procedure of Hermite interpolation of higher order based on the zeros of Jacobi polynomials plus the endpoints ±1 and prove that such a procedure can always well approximate a function and its derivatives simultaneously in uniform norm.