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

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 suggest that the Lebesgue constant behaves similarly for interpolation at Chebyshev as well as logarithmically distributed nodes.     

On the Lebesgue constant of barycentric rational interpolation at equidistant nodes

Recent results reveal that the family of barycentric rational interpolants introduced by Floater and Hormann is very well-suited for the approximation of functions as well as their derivatives, integrals and primitives. Especially in the case of equidistant interpolation nodes, these infinitely smooth interpolants offer a much better choice than their polynomial analogue. A natural and important question concerns the condition of this rational approximation method. In this paper we extend a recent study of the Lebesgue function and constant associated with Berrut’s rational interpolant at equidistant nodes to the family of Floater–Hormann interpolants, which includes the former as a special case.     

Evaluation of Lebesgue constant

The Lebesgue constant associated with interpolation at nodes U is evaluated in this note. The main result is an improvement on the estimate obtained by Brutman.     

Lebesgue constant for Lagrange interpolation on equidistant nodes

Properties of Lebesgue function for Lagrange interpolation on equidistant nodes are investigated. It is proved that Lebesgue function can be formulated both in terms of a hypergeometric function 2F1 and Jacobt polynomials. Moreover, an integral expression of Lebesgue function is also obtained and the asymptotic behavior of Lebesgue constant is studied.     

Polynomial Approximation and Interpolation on the Real Line with Respect to General Classes of Weights

Results on weighted polynomial approximation and interpolation with respect to Freud weights are extended to a more general class of weights. Among others, weights having zeros (Freud-Jacobi type weights) are considered, and a system of nodes is constructed for which the weighted Lebesgue constant of Lagrange interpolation is of optimal order.     

A Lobatto interpolation grid over the triangle

^** Email: m.blyth{at}uea.ac.uk^*** Email: cpozrikidis{at}ucsd.edu A sequence of increasingly refined interpolation grids overthe triangle is proposed, with the goal of achieving uniformconvergence and ensuring high interpolation accuracy. The numberof interpolation nodes, N, corresponds to a complete mth-orderpolynomial expansion with respect to the triangle barycentriccoordinates, which arises by the horizontal truncation of thePascal triangle. The proposed grid is generated by deployingLobatto interpolation nodes along the three edges of the triangle,and then computing interior nodes by averaged intersectionsto achieve three-fold rotational symmetry. Numerical computationsshow that the Lebesgue constant and interpolation accuracy ofthe proposed grid compares favorably with those of the best-knowngrids consisting of the Fekete points. Integration weights correspondingto the set of Lobatto triangle base points are tabulated.     

Sharp Lebesgue constants for bounded cubic interpolation ℒ-splines

We construct the Lebesgue function and find sharp Lebesgue constants for bounded cubic interpolation ℒ-splines with equally spaced interpolation nodes and discontinuities of the second derivative chosen so that the cubic ℒ-splines satisfy a certain extremal property with respect to the functions under interpolation.     

Exact Lebesgue constants for interpolatory ℒ-splines of third order

In this paper, we obtain the Lebesgue constants for interpolatory ?-splines of third order with uniform nodes, i.e., the norms of interpolation operators from C to C describing the process of interpolation of continuous bounded and continuous periodic functions by ?-splines of third order with uniform nodes on the real line. As a corollary, we obtain exact Lebesgue constants for interpolatory polynomial parabolic splines with uniform nodes.     

Mean Convergence of Extended Lagrange Interpolation with Freud Weights

Let W := exp(-Q), where Q is of smooth polynomial growth at , for example Q(x) = |x|, > 1. We call W² a Freud weight. The mean convergence of Lagrange interpolation at the zeros of the orthonormal polynomials associated with the Freud weight WW² has been studied by several authors, as has the Lebesgue function of Lagrange interpolation. J. Szabados had the idea to add two additional points of interpolation, thereby reducing the Lebesgue constant to grow no faster than log n. In this paper, we show that mean convergence of Lagrange interpolation at this extended set of nodes displays a similar advantage over merely using the zeros of the orthogonal polynomials.     

Pseudospectra of Matrix Polynomials that Are Expressed in Alternative Bases

Spectra and pseudospectra of matrix polynomials are of interest in geometric intersection problems, vibration problems, and analysis of dynamical systems. In this note we consider the effect of the choice of polynomial basis on the pseudospectrum and on the conditioning of the spectrum of regular matrix polynomials. In particular, we consider the direct use of the Lagrange basis on distinct interpolation nodes, and give a geometric characterization of “good” nodes. We also give some tools for computation of roots at infinity via a new, natural, reversal. The principal achievement of the paper is to connect pseudospectra to the well-established theory of Lebesgue functions and Lebesgue constants, by separating the influence of the scalar basis from the natural scale of the matrix polynomial, which allows many results from interpolation theory to be applied. This work was partially funded by the Natural Sciences and Engineering Research Council of Canada, and by the MITACS Network of Centres of Excellence.     

修正的 Thiele-Werner型有理插值

Through adjusting the order of interpolation nodes, we gave a kind of modified Thiele-Werner rational interpolation. This interpolation method not only avoids the infinite value of inverse differences in constructing the Thiele continued fraction interpolation, but also simplifies the interpolating polynomial coefficients with constant coefficients in the Thiele-Werner rational interpolation. Unattainable points and determinantal expression for this interpolation are considered. As an extension, some bivariate analogy is also discussed and numerical examples are given to show the validness of this method.     

Interpolation by Polynomials and Radial Basis Functions on Spheres

The paper obtains error estimates for approximation by radial basis functions on the sphere. The approximations are generated by interpolation at scattered points on the sphere. The estimate is given in terms of the appropriate power of the fill distance for the interpolation points, in a similar manner to the estimates for interpolation in Euclidean space. A fundamental ingredient of our work is an estimate for the Lebesgue constant associated with certain interpolation processes by spherical harmonics. These interpolation processes take place in ``spherical caps' whose size is controlled by the fill distance, and the important aim is to keep the relevant Lebesgue constant bounded. This result seems to us to be of independent interest. March 27, 1997. Dates revised: March 19, 1998; August 5, 1999. Date accepted: December 15, 1999.     

On the Lebesgue constant of Leja sequences for the unit disk and its applications to multivariate interpolation

We estimate the growth of the Lebesgue constant of any Leja sequence for the unit disk. The main application is the construction of new multivariate interpolation points in a polydisk (and in the Cartesian product of many plane compact sets) whose Lebesgue constant grows (at most) like a polynomial.     

Lebesgue functions corresponding to a family of Lagrange interpolation polynomials

In this paper we obtain various explicit forms of the Lebesgue function corresponding to a family of Lagrange interpolation polynomials defined at an even number of nodes. We study these forms by using the derivatives up to the second order inclusive. We estimate exact values of Lebesgue constants for this family from below and above in terms of known parameters. In a particular case we obtain new convenient formulas for calculating these estimates.     

三维空间中的有理样条插值

对三维空间某个多面体区域的四面体剖分，通过在每个四面体胞腔的棱和顶点设置适当的插值结点．本文给出了（１，１）型Ｃ⁰及Ｃ¹光滑的非奇异有理样条存在的充分必要条件．     

Polyharmonic and Related Kernels on Manifolds: Interpolation and Approximation

This article is devoted to developing a theory for effective kernel interpolation and approximation in a general setting. For a wide class of compact, connected C ^∞ Riemannian manifolds, including the important cases of spheres and SO(3), and using techniques involving differential geometry and Lie groups, we establish that the kernels obtained as fundamental solutions of certain partial differential operators generate Lagrange functions that are uniformly bounded and decay away from their center at an algebraic rate, and in certain cases, an exponential rate. An immediate corollary is that the corresponding Lebesgue constants for interpolation as well as for L ₂ minimization are uniformly bounded with a constant whose only dependence on the set of data sites is reflected in the mesh ratio, which measures the uniformity of the data. The kernels considered here include the restricted surface splines on spheres, as well as surface splines for SO(3), both of which have elementary closed-form representations that are computationally implementable. In addition to obtaining bounded Lebesgue constants in this setting, we also establish a “zeros lemma” for domains on compact Riemannian manifolds—one that holds in as much generality as the corresponding Euclidean zeros lemma (on Lipschitz domains satisfying interior cone conditions) with constants that clearly demonstrate the influence of the geometry of the boundary (via cone parameters) as well as that of the Riemannian metric.     

On the generation of symmetric Lebesgue-like points in the triangle

We compute point sets on the triangle that have low Lebesgue constant, with sixfold symmetries and Gauss–Legendre–Lobatto distribution on the sides, up to interpolation degree 18. Such points have the best Lebesgue constants among the families of symmetric points used so far in the framework of triangular spectral elements.     

A complete description of the Lebesgue functions for classical lagrange interpolation polynomials

Explicit forms of Lebesgue functions are not described in the mathematical literature by now. This issue is related to the problem of reducing sums of modules of fundamental polynomials or the corresponding Dirichlet kernels. That is why the complete study of graphs of Lebesgue functions remains a complicated topical problem in the theory of approximation. In this paper we solve the mentioned problems both for odd and even numbers of interpolation nodes. We find explicit forms of Lebesgue functions and study them by means of the differential calculus. All mentioned forms are new.     

Asymptotic formulas for Lebesgue functions corresponding to the family of Lagrange interpolation polynomials

Mathematical Notes - The asymptotic behavior of Lebesgue functions of trigonometric Lagrange interpolation polynomials constructed on an even number of nodes is studied. For these functions,...     

Lower bound for the Lebesgue function of an interpolation process with algebraic polynomials on equidistant nodes of a simplex

For an interpolation process with algebraic polynomials of degree n on equidistant nodes of an m-simplex for m ≥ 2, we obtain a pointwise lower bound for the Lebesgue function similar to the well-known estimate for interpolation on a closed interval.