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.     

Complex versus real orthogonal polynomials of two variables

Orthogonal polynomials of two real variables can often be represented in complex variables. We explore the connection between the two types of representations and study the structural relations of complex orthogonal polynomials. The complex Hermite orthogonal polynomials and the disk polynomials are used as illustrating examples.     

On the divided differences of the remainder in polynomial interpolation

We present formulas for the divided differences of the remainder of the interpolation polynomial that include some recent interesting formulas as special cases.     

Stieltjes polynomials and Lagrange interpolation

Bounds are proved for the Stieltjes polynomial , and lower bounds are proved for the distances of consecutive zeros of the Stieltjes polynomials and the Legendre polynomials . This sharpens a known interlacing result of Szegö. As a byproduct, bounds are obtained for the Geronimus polynomials . Applying these results, convergence theorems are proved for the Lagrange interpolation process with respect to the zeros of , and for the extended Lagrange interpolation process with respect to the zeros of in the uniform and weighted norms. The corresponding Lebesgue constants are of optimal order.

     

Optimal interpolation of convergent algebraic series

Let , –1<x ₁<...<x _n <1. Denote , t∈(–1,1). Given a function f∈W we try to recover f(ζ) at fixed point ζ∈(–1,1) by an algorithm A on the basis of the information f(x ₁),...,f(x _n ). We find the intrinsic error of recovery . This work is supported by RFBR (grant 07-01-00167-a and grant 06-01-00003).     

On multivariate polynomial interpolation

We provide a map which associates each finite set in complexs-space with a polynomial space from which interpolation to arbitrary data given at the points in is possible and uniquely so. Among all polynomial spacesQ from which interpolation at is uniquely possible, our is of smallest degree. It is alsoD- and scale-invariant. Our map is monotone, thus providing a Newton form for the resulting interpolant. Our map is also continuous within reason, allowing us to interpret certain cases of coalescence as Hermite interpolation. In fact, our map can be extended to the case where, with eachgq, there is associated a polynomial space P, and, for given smoothf, a polynomialqQ is sought for which

On the Sauer-Xu formula for the error in multivariate polynomial interpolation

Use of a new notion of multivariate divided difference leads to a short proof of a formula by Sauer and Xu for the error in interpolation, by polynomials of total degree in variables, at a `correct' point set.

     

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.     

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.     

On the poisedness of Bojanov–Xu interpolation

In this paper, we consider the bivariate Hermite interpolation introduced by Bojanov and Xu [SIAM J. Numer. Anal. 39(5) (2002) 1780–1793]. The nodes of the interpolation with Π_2k-δ, where δ=0 or 1, are the intersection points of 2k+1 distinct rays from the origin with a multiset of k+1-δ concentric circles. Parameters are the values and successive radial derivatives, whenever the corresponding circle is multiple. The poisedness of this interpolation was proved only for the set of equidistant rays [Bojanov and Xu, 2002] and its counterparts with other conic sections [Hakopian and Ismail, East J. Approx. 9 (2003) 251–267]. We show that the poisedness of this (k+1-δ)(2k+1) dimensional Hermite interpolation problem is equivalent to the poisedness of certain 2k+1 dimensional Lagrange interpolation problems. Then the poisedness of Bojanov–Xu interpolation for a wide family of sets of rays satisfying some simple conditions is established. Our results hold also with above circles replaced by ellipses, hyperbolas, and pairs of parallel lines.Next a conjecture [Hakopian and Ismail, J. Approx. Theory 116 (2002) 76–99] concerning a poisedness relation between the Bojanov–Xu interpolation, with set of rays symmetric about x-axis, and certain univariate lacunary interpolations is established. At the end the poisedness for a wide class of lacunary interpolations is obtained.     

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.     

On specific stability bounds for linear multiresolution schemes based on piecewise polynomial Lagrange interpolation

The Deslauriers-Dubuc symmetric interpolation process can be considered as an interpolatory prediction scheme within Harten's framework. In this paper we express the Deslauriers-Dubuc prediction operator as a combination of either second order or first order differences. Through a detailed analysis of certain contractivity properties, we arrive to specific l_∞-stability bounds for the multiresolution transform. A variety of tests indicate that these l_∞ bounds are closer to numerical estimates than those obtained with other approaches.     

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.     

Error inequalities for quintic and biquintic discrete Hermite interpolation

In this paper we shall develop a class of discrete Hermite interpolates in one and two independent variables. Further, we offer explicit error bounds in ?_∞ norm for the quintic and biquintic discrete Hermite interpolates. Some numerical examples are included to illustrate the results obtained.     

Regular points for Lagrange interpolation on the unit disk

A set of points on the unit disk of the Euclidean plane is given, which admits unique Lagrange interpolation. The points have rotational symmetry and they form an example of natural lattices of Chung and Yao [2]. Properties of Lagrange interpolation with respect to these points are studied.Work done when visiting the University of Oregon at Eugene, Oregon.Supported by National Science Foundation under Grant No. 9302721.     

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.     

Degenerate polynomial patches of degree 11 for almostGC 2 interpolation over triangles

We consider the problem of interpolating scattered data in ³ by analmost geometrically smoothGC ² surface, where almostGC ² meansGC ² except in a finite number of points (the vertices), where the surface isGC ¹. A local method is proposed, based on employing so-called degenerate triangular Bernstein-Bézier patches. We give an analysis of quintic patches forGC ¹ and patches of degree eleven for almostGC ² interpolation.