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.     

A survey of truncation error analysis for Padé and continued fraction approximants

To compute the value of a functionf(z) in the complex domain by means of a converging sequence of rational approximants {f _n(z)} of a continued fraction and/or Padé table, it is essential to have sharp estimates of the truncation error ¦f(z)–f _n(z)¦. This paper is an expository survey of constructive methods for obtaining such truncation error bounds. For most cases dealt with, {f _n(z)} is the sequence of approximants of a continued fractoin, and eachf _n(z) is a (1-point or 2-point) Padé approximant. To provide a common framework that applies to rational approximantf _n(z) that may or may not be successive approximants of a continued fraction, we introduce linear fractional approximant sequences (LFASs). Truncation error bounds are included for a large number of classes of LFASs, most of which contain representations of important functions and constants used in mathematics, statistics, engineering and the physical sciences. An extensive bibliography is given at the end of the paper.Research supported in part by the U.S. National Science Foundation under Grants INT-9113400 and DMS-9302584.     

A note on an error estimate for least squares approximation

An asymptotic expansion is obtained which provides upper and lower bounds for the error of the bestL ₂ polynomial approximation of degreen forx ⁿ⁺¹ on [–1, 1]. Because the expansion proceeds in only even powers of the reciprocal of the large variable, and the error made by truncating the expansion is numerically less than, and has the same sign as the first neglected term, very good bounds can be obtained. Via a result of Phillips, these results can be extended fromx ⁿ⁺¹ to anyfC ⁿ⁺¹[–1, 1], provided upper and lower bounds for the modulus off ⁽ⁿ⁺¹⁾ are available.     

Asymptotic expansion formula for Bernstein polynomials defined on a simplex

In this paper we give a complete expansion formula for Bernstein polynomials defined on ans-dimensional simplex. This expansion for a smooth functionf represents the Bernstein polynomialB _n (f) as a combination of derivatives off plus an error term of orderO(n^–s ).Communicated by Wolfgang Dahmen.     

Hermite-Fejer type interpolation of higher order

The paper introduces Hermite-Fejér type (Hermite type) interpolation of higher order denoted by S _mn(f)(S* _mm(f)), and gives some basic properties including expression formulas, convergence relationship between S _mn(f) and H _mn(f) (Hermite-Fejér interpolation of higher order), and the saturation of S _mn(f). Supported by the Science Foundation of Shanxi Province for Returned Scholars.     

Integral error formulæ for the scale of mean value interpolations which includes Kergin and Hakopian interpolation

Summary. In this paper, we provide an integral error formula for a certain scale of mean value interpolations which includes the multivariate polynomial interpolation schemes of Kergin and Hakopian. This formula involves only derivatives of order one higher than the degree of the interpolating polynomial space, and from it we can obtain sharp -estimates. These -estimates are precisely those that numerical analysts want, to guarantee that a scheme based on such an interpolation has the maximum possible order. Received July 11, 1994 / Revised version received February 12, 1996     

Multipoint Taylor formulæ

Summary. The main result of this paper is an abstract version of the Kowalewski–Ciarlet–Wagschal multipoint Taylor formula for representing the pointwise error in multivariate Lagrange interpolation. Several applications of this result are given in the paper. The most important of these is the construction of a multipoint Taylor error formula for a general finite element, together with the corresponding –error bounds. Another application is the construction of a family of error formul? for linear interpolation (indexed by real measures of unit mass) which includes some recently obtained formul?. It is also shown how the problem of constructing an error formula for Lagrange interpolation from a D–invariant space of polynomials with the property that it involves only derivatives which annihilate the interpolating space can be reduced to the problem of finding such a formula for a ‘simpler’ one–point interpolation map. Received March 29, 1996 / Revised version received November 22, 1996     

An extension of a bound for functions in Sobolev spaces,with applications to (<Emphasis Type="Italic">m</Emphasis>, <Emphasis Type="Italic">s</Emphasis>)-spline interpolation and smoothing

Given a function f on a bounded open subset Ω of with a Lipschitz-continuous boundary, we obtain a Sobolev bound involving the values of f at finitely many points of . This result improves previous ones due to Narcowich et al. (Math Comp 74, 743–763, 2005), and Wendland and Rieger (Numer Math 101, 643–662, 2005). We then apply the Sobolev bound to derive error estimates for interpolating and smoothing (m, s)-splines. In the case of smoothing, noisy data as well as exact data are considered.     

Greedy Algorithm and m -Term Trigonometric Approximation

We study the following nonlinear method of approximation by trigonometric polynomials in this paper. For a periodic function f we take as an approximant a trigonometric polynomial of the form , where is a set of cardinality m containing the indices of the m biggest (in absolute value) Fourier coefficients of function f . We compare the efficiency of this method with the best m -term trigonometric approximation both for individual functions and for some function classes. It turns out that the operator G _m provides the optimal (in the sense of order) error of m -term trigonometric approximation in the L _p -norm for many classes. September 23, 1996. Date revised: February 3, 1997.     

Convex polynomial and spline approximation inL p,O<p<∞

We prove that a convex functionf ∈ L _p[−1, 1], 0<p<∞, can be approximated by convex polynomials with an error not exceeding Cω ₃ ^ϕ (f,1/n)_p where ω ₃ ^ϕ (f,·) is the Ditzian-Totik modulus of smoothness of order three off. We are thus filling the gap between previously known estimates involving ω ₃ ^ϕ (f,1/n)_p, and the impossibility of having such estimates involving ω₄. We also give similar estimates for the approximation off by convexC ⁰ andC ¹ piecewise quadratics as well as convexC ² piecewise cubic polynomials. Communicated by Dietrich Braess     

A family of Hermite interpolants by bisection algorithms

A two point subdivision scheme with two parameters is proposed to draw curves corresponding to functions that satisfy Hermite conditions on [a, b]. We build two functionsf andf ¹ on dyadic numbers and for some values of the parameters,f is in ¹ withf ¹=f. Examples are provided which show how different the curves can be.     

The approximation by q-Bernstein polynomials in the case q ↓ 1

Let B_n (f, q; x), n=1, 2, ... , 0 < q < ∞, be the q-Bernstein polynomials of a function f, B_n (f, 1; x) being the classical Bernstein polynomials. It is proved that, in general, {B_n (f, q_n; x)} with q_n ↓ 1 is not an approximating sequence for f ∈C[0, 1], in contrast to the standard case q_n ↓ 1. At the same time, there exists a sequence 0 < δ_n ↓ 0 such that the condition implies the approximation of f by {B_n (f, q_n; x)} for all f ∈C[0, 1]. Received: 15 March 2005     

Interpolating them-th power ofx at the zeros of then-th Chebyshev-polynomial yields an almost best Chebyshev-approximation

LetX={x ₁,x ₂,..., _n }I=[–1, 1] and . ForfC ₁(I) definef* byf–p _f =f*, wherep _f denotes the interpolation-polynomial off with respect toX. We state some properties of the operatorf f*. In particular, we treat the case whereX consists of the zeros of the Chebyshev polynomialT _n (x) and obtain x ^m –p _x ^m8eE _n–1(x ^m ), whereE _n–1(f) denotes the sup-norm distance fromf to the polynomials of degree less thann. Finally we state a lower estimate forE _n (f) that omits theassumptionf ⁽ⁿ⁺¹⁾>0 in a similar estimate of Meinardus.     

Simultaneous approximation by Bernstein operators in Hölder norms

In this paper we discuss approximation of continuous functions f on [0, 1] in Hölder norms including simultaneous approximation of derivatives of f.     

Saturation of Kantorovich type operators

It is proved that an integrable functionf can be approximated by the Kantorovich type modification of the Szász—Mirakjan and Baskakov operators inL ¹ metric in the optimal order {n ^–1} if and only if ² f is of bounded variation where and , respectively.     

On the construction of multi-dimensional embedded cubature formulae

Summary In order to compute an integralI[f], one needs at least two cubature formulaeQ _j ,j{1, 2}. |Q ₁[f]–Q ₂[f]| can be used as an error estimate for the less precise cubature formula. In order to reduce the amount of work, one can try to reuse some of the function evaluations needed forQ ₁, inQ ₂. The easiest way to construct embedded cubature formulae is: start with a high degree formulaQ ₁, drop (at least) one knot and calculate the weights such that the new formulaQ ₂ is exact for as much monomials as possible. We describe how such embedded formulae with positive weights can be found. The disadvantage of such embedded cubature formulae is that there is in general a large difference in the degree of exactness of the two formulae. In this paper we will explain how the high degree formula can be chosen to obtain an embedded pair of cubature formulae of degrees 2m+1/2m–1. The method works for all regions ⁿ ,n2. We will also show the influence of structure on the cubature formulae.     

Convex polynomial and spline approximation in C[−1, 1]

We prove that a convex functionf C[–1, 1] can be approximated by convex polynomialsp _n of degreen at the rate of ₃(f, 1/n). We show this by proving that the error in approximatingf by C² convex cubic splines withn knots is bounded by ₃(f, 1/n) and that such a spline approximant has anL third derivative which is bounded by n³₃(f, 1/n). Also we prove that iff C²[–1, 1], then it is approximable at the rate ofn ^–2 (f, 1/n) and the two estimates yield the desired result.Communicated by Ronald A. DeVore.     

Error bounds for a near-minimax approximation

It is known that a near minimax polynomial approximation tof C [–1, 1] is provided by a finite carrier projectionM _n fromC[–1, 1] onto the subspace of all polynomials of degree n, such thatM _nf is a weighted least squares approximation off on the set consisting of the extreme points of the Chebyshev polynomialT _{2n + 1}. In this paper, upper bounds for the error f–M _n f are given in terms of divided differences.     

Bivariate spline interpolation with optimal approximation order

Let Δ be a triangulation of some polygonal domain Ω ⊂ R² and let S_q^r(Δ) denote the space of all bivariate polynomial splines of smoothness r and degree q with respect to Δ. We develop the first Hermite-type interpolation scheme for S _q ^r (Δ), q ≥ 3r + 2, whose approximation error is bounded above by Kh ^q +1, where h is the maximal diameter of the triangles in Δ, and the constant K only depends on the smallest angle of the triangulation and is independent of near-degenerate edges and near-singular vertices. Moreover, the fundamental functions of our scheme are minimally supported and form a locally linearly independent basis for a superspline subspace of S _q ^r (Δ). This shows that the optimal approximation order can be achieved by using minimally supported splines. Our method of proof is completely different from the quasi-interpolation techniques for the study of the approximation power of bivariate splines developed in [7] and [18].     

EfficientL 2 approximation by splines

LetS _N ^k (t) be the linear space ofk-th order splines on [0, 1] having the simple knotst _i determined from a fixed functiont by the rulet _i=t(i/N). In this paper we introduce sequences of operators {Q _N } _N =1 fromC ^k [0, 1] toS _N ^k (t) which are computationally simple and which, asN, give essentially the best possible approximations tof and its firstk–1 derivatives, in the norm ofL ₂[0, 1]. Precisely, we show thatN ^k–1((f–Q _N f) ⁱ –dist₂(f ⁽¹⁾,S _N ^k–1 (t)))0 fori=0, 1, ...,k–1. Several numerical examples are given.The research of this author was partially supported by the National Science Foundation under Grant MCS-77-02464The research of this author was partially supported by the U.S. Army Reesearch Office under Grant No. DAHC04-75-G-0816