Lagrange Interpolation in Weighted Besov Spaces

The behavior of the Lagrange polynomial L _m (w,f) , based on the zeros of the orthogonal polynomials, is studied in some weighted Besov spaces B ^p _r,q (u) . It is proved that L _m (w) is a uniformly bounded map under suitable conditions on the weight functions and the parameters p , r , and q . December 11, 1996. Date revised: October 29, 1997. Date accepted: June 15, 1998.     

On Rational Interpolation to |x|

We consider Newman-type rational interpolation to |x| induced by arbitrary sets of interpolation nodes, and we show that under mild restrictions on the location of the interpolation nodes, the corresponding sequence of rational interpolants converges to |x|. Date received: August 18, 1995. Date revised: January 10, 1996.     

On the Zero-Divergence of Equidistant Lagrange Interpolation

 In 1942, P. Szász published the surprising result that if a function f is of bounded variation on [−1, 1] and continuous at 0 then the sequence of the equidistant Lagrange interpolation polynomials converges at 0 to . In the present note we give a construction of a function continuous on [−1, 1] whose Lagrange polynomials diverge at 0. Moreover, we show that the rate of divergence attains almost the maximal possible rate. (Received 2 February 2000)     

On weighted Lebesgue function type sums

Let I be a finite or infinite interval and dμ a measure on I. Assume that the weight function w(x)>0, w^′(x) exists, and the function w^′(x)/w(x) is non-increasing on I. Denote by ℓ_k's the fundamental polynomials of Lagrange interpolation on a set of nodes x₁<x₂<<x_n in I. The weighted Lebesgue function type sum for 1≤i<j≤n and s≥1 is defined by

In this paper the exact lower bounds of S_n(x) on a “big set” of I and are obtained. Some applications are also given.     

Weighted Polynomial Approximation for Convex External Fields

It is proven that if Q is convex and w(x)= exp(-Q(x)) is the corresponding weight, then every continuous function that vanishes outside the support of the extremal measure associated with w can be uniformly approximated by weighted polynomials of the form w ⁿ P _n . This solves a problem of P. Borwein and E. B. Saff. Actually, a similar result is true locally for any parts of the extremal support where Q is convex. February 10, 1998. Date revised: July 23, 1998. Date accepted: August 17, 1998.     

Approximation and Interpolation from Modules

Abstract

A result about simultaneous approximation and interpolation from modules of weighted spaces is established. As a consequence, it is applied to certain polynomial algebras of the space of continuous bounded vector-valued functions equipped with the strict topology.     

On the Power of Standard Information for Weighted Approximation

We study weighted approximation of multivariate functions for classes of standard and linear information in the worst case and average case settings. Under natural assumptions, we show a relation between n th minimal errors for these two classes of information. This relation enables us to infer convergence and error bounds for standard information, as well as the equivalence of tractability and strong tractability for the two classes. April 11, 2001. Final version received: May 29, 2001.     

L p -Error Bounds for Hermite Interpolation and the Associated Wirtinger Inequalities

The B-spline representation for divided differences is used, for the first time, to provide L _p -bounds for the error in Hermite interpolation, and its derivatives, thereby simplifying and improving the results to be found in the extensive literature on the problem. These bounds are equivalent to certain Wirtinger inequalities. The major result is the inequality where H_Θ f is the Hermite interpolant to f at the multiset of n points Θ, and is the diameter of . This inequality significantly improves upon Beesack's inequality, on which almost all the bounds given over the last 30 years have been based. Date received: June 24, 1994 Date revised: February 4, 1996.     

Asymptotic Distribution of Nodes for Near-Optimal Polynomial Interpolation on Certain Curves in R 2

Let E\subset \Bbb R ^s be compact and let d _n ^E denote the dimension of the space of polynomials of degree at most n in s variables restricted to E . We introduce the notion of an asymptotic interpolation measure (AIM). Such a measure, if it exists , describes the asymptotic behavior of any scheme τ _n ={ \bf x _k,n } _k=1 ^dnE , n=1,2,\ldots , of nodes for multivariate polynomial interpolation for which the norms of the corresponding interpolation operators do not grow geometrically large with n . We demonstrate the existence of AIMs for the finite union of compact subsets of certain algebraic curves in R ² . It turns out that the theory of logarithmic potentials with external fields plays a useful role in the investigation. Furthermore, for the sets mentioned above, we give a computationally simple construction for ``good' interpolation schemes. November 9, 2000. Date revised: August 4, 2001. Date accepted: September 14, 2001.     

Interpolation of medians

We study the median of a continuous function on an interval and show that for certain spaces of functions there is a unique function in the space whose medians on given intervals take given values. This revised version was published online in June 2006 with corrections to the Cover Date.     

Weighted Polynomial Inequalities with Doubling and A ∞ Weights

We consider weighted inequalities such as Bernstein, Nikolskii, Remez, etc., inequalities under minimal assumptions on the weights. It turns out that in most cases this mimimal assumption is the doubling condition. Sometimes, however, as for the Remez and Nikolskii inequalities, one needs the slightly stronger A _∈ fty condition. We shall consider both the trigonometric and the algebraic cases. August 20, 1997. Date revised: April 19, 1998. Date accepted: May 26, 1998.     

Balayage Properties Related to Rational Interpolation

Ambroladze and Wallin have posed several problems, about balayage of measures, which arose from work on approximation by polynomial and rational interpolation in the complex plane. These problems concern the possible coincidence of measures swept onto a Jordan curve from the inner and outer domains. This paper describes when the desired balayage properties hold.     

Orthonormality of Cardinal Chebyshev B-Spline Bases in Weighted Sobolev Spaces

Extending a recent result of Ulrich Reif on cardinal polynomial B-splines, we show that the cardinal Chebyshev B-spline basis associated with a linear differential operator with constant real coefficients is orthonormal with respect to a unique weighted Sobolev-type inner product.     

Weighted Polynomial Approximation in the Complex Plane

Given a pair (G,W) of an open bounded set G in the complex plane and a weight function W(z) which is analytic and different from zero in G , we consider the problem of the locally uniform approximation of any function f(z) , which is analytic in G , by weighted polynomials of the form {W ⁿ (z)P _n (z) } ^$\infinity$ _n=0 , where deg P_n n. The main result of this paper is a necessary and sufficient condition for such an approximation to be valid. We also consider a number of applications of this result to various classical weights, which give explicit criteria for these weighted approximations. May 1, 1996. Date revised: October 8, 1996.     

On Divergence of Hermite—Fejér Interpolation to f(z)=z in the Complex Plane

We show that for a broad class of interpolatory matrices on [-1,1] the sequence of polynomials induced by Hermite—Fejér interpolation to f(z)=z diverges everywhere in the complex plane outside the interval of interpolation [-1,1] . This result is in striking contrast to the behavior of the Lagrange interpolating polynomials. June 15, 1998. Date accepted: January 26, 1999.     

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 Divergence of Polynomial Interpolation in the Complex Plane

We extend the results in [1] and [2] from the divergence of Hermite—Fejér interpolation in the complex plane to the divergence of arbitrary polynomial interpolation in the complex plane. In particular, we prove the following theorem: Let \D _n =-1≤ t ₁ ⁽ⁿ⁾ <⋅s<t _n ⁽ⁿ⁾ <1 . Let \v _k ⁽ⁿ⁾ be polynomials of arbitrary degree such that \v _k ⁽ⁿ⁾ (t _j ⁽ⁿ⁾ )=\d _kj . Then the Lebesgue function tends to infinity at every complex neighborhood of some point in [-1,1] . March 23, 2000. Date revised: September 28, 2000. Date accepted: October 10, 2000.     

On the Negative Extremums of Fundamental Functions of Lagrange Interpolation Based on Chebyshev Nodes

In this paper, we investigate the negative extremums of fundamental functions of Lagrange interpolation based on Chebyshev nodes. Moreover, we establish some companion results to the theorem of J. Szabados on the positive extremum.     

On Converse Marcinkiewicz—Zygmund Inequalities in L p ,p>1

We obtain converse Marcinkiewicz—Zygmund inequalities such as for polynomials P of degree ≤ n-1 , under general conditions on the points {t _j } ⁿ _j=1 and on the function ν . The weights {μ _j } ⁿ _j=1 are appropriately chosen. We illustrate the results by applying them to extended Lagrange interpolation for exponential weights on [-1,1] . December 3, 1997. Date revised: December 7, 1998. Date accepted: January 8, 1999.     

On Stable Local Bases for Bivariate Polynomial Spline Spaces

Stable locally supported bases are constructed for the spaces \cal S _d ^r (\triangle) of polynomial splines of degree d≥ 3r+2 and smoothness r defined on triangulations \triangle , as well as for various superspline subspaces. In addition, we show that for r≥ 1 , in general, it is impossible to construct bases which are simultaneously stable and locally linearly independent. February 2, 2000. Date revised: November 27, 2000. Date accepted: March 7, 2001.