On Genuine Bernstein–Durrmeyer Operators

We continue the studies on the so–called genuine Bernstein–Durrmeyer operators U _n by establishing a recurrence formula for the moments and by investigating the semigroup T(t) approximated by U _n . Moreover, for sufficiently smooth functions the degree of this convergence is estimated. We also determine the eigenstructure of U _n , compute the moments of T(t) and establish asymptotic formulas. Received: January 26, 2007.     

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     

Local approximation results for Szász-Mirakjan type operators

In this study, motivating our earlier work [O. Duman and M.A. ?zarslan, Szász-Mirakjan type operators providing a better error estimation. Appl. Math. Lett. 20, 1184–1188 (2007)], we investigate the local approximation properties of Szász-Mirakjan type operators. The second modulus of smoothness and Petree’s K-functional are considered in proving our results. Received: 17 September 2007     

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.     

Integral-type operators on continuous function spaces on the real line

In this paper we introduce some new sequences of positive linear operators, acting on a sufficiently large space of continuous functions on the real line, which generalize Gauss–Weierstrass operators.We study their approximation properties and prove an asymptotic formula that relates such operators to a second order elliptic differential operator of the form Lu?αu^′′+βu^′+γu.Shape-preserving and regularity properties are also investigated.     

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.     

The uniform norm of hyperinterpolation on the unit sphere in an arbitrary number of dimensions

In this paper we study the order of growth of the uniform norm of the hyperinterpolation operator on the unit sphere S ^r−1 ⊂ R^r. The hyperinterpolation approximation L _n ƒ, where ƒ ∈ C(S ^r −1), is derived from the exact L ² orthogonal projection Π ƒ onto the space P _n ^r (S ^r −1) of spherical polynomials of degree n or less, with the Fourier coefficients approximated by a positive weight quadrature rule that integrates exactly all polynomials of degree ≤ 2n. We extend to arbitrary r the recent r = 3 result of Sloan and Womersley [9], by proving that under an additional “quadrature regularity” assumption on the quadrature rule, the order of growth of the uniform norm of the hyperinterpolation operator on the unit sphere is O(n ^r /2−1), which is the same as that of the orthogonal projection Π_n, and best possible among all linear projections onto P _n ^r (S ^r −1).     

Uniform rational approximation of functions with first derivative in the real hardy space ReH 1

The main result proved in the paper is: iff is absolutely continuous in (–, ) andf' is in the real Hardy space ReH ¹, then for everyn1, whereR _n(f) is the best uniform approximation off by rational functions of degreen. This estimate together with the corresponding inverse estimate of V. Russak [15] provides a characterization of uniform rational approximation.Communicated by Ronald A. DeVore.     

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.     

Regarding thep-norms of radial basis interpolation matrices

A radial basis function approximation has the form where:R ^d R is some given (usually radially symmetric) function, (y _j ) ₁ ⁿ are real coefficients, and the centers (x _j ) ₁ ⁿ are points inR ^d . For a wide class of functions , it is known that the interpolation matrixA=((x _j –x _k )) _j,k=1 ⁿ is invertible. Further, several recent papers have provided upper bounds on ||A ^–1||₂, where the points (x _j ) ₁ ⁿ satisfy the condition ||x _j –x _k ||₂,jk, for some positive constant . In this paper we calculate similar upper bounds on ||A ^–1||₂ forp1 which apply when decays sufficiently quickly andA is symmetric and positive definite. We include an application of this analysis to a preconditioning of the interpolation matrixA _n = ((j–k)) _j,k=1 ⁿ when (x)=(x ²+c ²)^1/2, the Hardy multiquadric. In particular, we show that sup _n ||A _n ^–1 || is finite. Furthermore, we find that the bi-infinite symmetric Toeplitz matrix enjoys the remarkable property that ||E ^–1|| _p = ||E ^–1||₂ for everyp1 when is a Gaussian. Indeed, we also show that this property persists for any function which is a tensor product of even, absolutely integrable Pólya frequency functions.Communicated by Charles Micchelli.     

Conditionally positive functions andp-norm distance matrices

In [4], deep results were obtained concerning the invertibility of matrices arising from radial basis function interpolation. In particular, the Euclidean distance matrix was shown to be invertible for distinct data. In this paper, we investigate the invertibility of distance matrices generated byp-norms. In particular, we show that, for anyp(1, 2), and for distinct pointsx ¹,,x ^{n d} , wheren andd may be any positive integers, with the proviso thatn2, the matrixA ^n×n defined by

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].     

Approximation in L p (R d ) from a space spanned by the scattered shifts of a radial basis function

A new multivariate approximation scheme on R ^d using scattered translates of the “shifted” surface spline function is developed. The scheme is shown to provide spectral L _p -approximation orders with 1 ≤ p ≤ ∞, i.e., approximation orders that depend on the smoothness of the approximands. In addition, it applies to noisy data as well as noiseless data. A numerical example is presented with a comparison between the new scheme and the surface spline interpolation method.     

Shape-preserving approximation inL p

We prove a direct theorem for shape preservingL _p -approximation, 0p, in terms of the classical modulus of smoothnessw ₂(f, t _p ¹ ). This theorem may be regarded as an extension toL _p of the well-known pointwise estimates of the Timan type and their shape-preserving variants of R. DeVore, D. Leviatan, and X. M. Yu. It leads to a characterization of monotone and convex functions in Lipschitz classes (and more general Besov spaces) in terms of their approximation by algebraic polynomials.Communicated by Ron DeVore.     

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     

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.     

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.     

Cardinal hermite interpolation with box splines II

Summary In the present work we extent the results in [RS] on CHIP, i.e. Cardinal Hermite Interpolation by the span of translates of directional derivatives of a box spline. These directional derivatives are that ones which define the type of the Hermite Interpolation. We admit here several (linearly independent) directions with multiplicities instead of one direction as in [RS]. Under the same assumptions on the smoothness of the box spline and its defining matrixT we can prove as in [RS]: CHIP has a system of fundamental solutions which are inL L ² together with its directional derivatives mentioned above. Moreover, for data sequences inl ^p ( ^d ), 1p2, there is a spline function inL ^p, 1/p+1/p=1, which solves CHIP.Research supported in part by NSERC Canada under Grant # A7687. This research was completed while this author was supported by a grant from the Deutscher Akademischer Austauschdienst     

Partition of unity and density: A counterexample

By providing a counterexample we show that there exists a shift-invariant spaceS generated by a piecewise linear function such that the union of the corresponding scaled spacesS ^h (h>0) is dense inC ₀(R ²) butS does not contain a stable and locally supported partition of unity. This settles a question raised by C. de Boor and R. DeVore a decade ago.     

From continuous to discrete Weyl-Heisenberg frames through sampling

In this article we consider the question when one can generate a Weyl- Heisenberg frame for l ² (ℤ) with shift parameters N, M ⁻¹ (integer N, M) by sampling a Weyl-Heisenberg frame for L ² (ℝ) with the same shift parameters at the integers. It is shown that this is possible when the window g ε L ² (ℝ) generating the Weyl-Heisenberg frame satisfies an appropriate regularity condition at the integers. When, in addition, the Tolimieri-Orr condition A is satisfied, the minimum energy dual window ^o γ ε L ² (ℝ) can be sampled as well, and the two sampled windows continue to be related by duality and minimality. The results of this article also provide a rigorous basis for the engineering practice of computing dual functions by writing the Wexler-Raz biorthogonality condition in the time-domain as a collection of decoupled linear systems involving samples of g and ^o γ as knowns and unknowns, respectively. We briefly indicate when and how one can generate a Weyl-Heisenberg frame for the space of K-periodic sequences, where K=LCM (N, M), by periodization of a Weyl-Heisenberg frame for ℓ ² ℤ with shift parameters N, M ⁻¹ .