Inequalities for generalized nonnegative polynomials

We define generalized polynomials as products of polynomials raised to positive real powers. The generalized degree can be defined in a natural way. We prove Markov-, Bernstein-, and Remez-type inequalities inL ^p (0p) and Nikolskii-type inequalities for such generalized polynomials. Our results extend the corresponding inequalities for ordinary polynomials.Communicated by George G. Lorentz.     

Markov-Bernstein-type inequalities for classes of polynomials with restricted zeros

We prove that an absolute constantc>0 exists such that

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.     

Asymptotics of polynomials orthogonal with respect to varying measures

Letd be a finite positive Borel measure on the interval [0, 2] such that >0 almost everywhere; andW _n be a sequence of polynomials, degW _n =n, whose zeros (w _n ,1,,w _n,n lie in [|z|1]. Let d _n <> for each_nN, whered _n =d/|W _n (e ⁱ )|². We consider the table of polynomials _n,m such that for each fixednN the system _n,m,mN, is orthonormal with respect tod _n . If

Explicit generalized zolotarev polynomials with complex coefficients

We give explicitly a class of polynomials with complex coefficients of degreen which deviate least from zero on [−1, 1] with respect to the max-norm among all polynomials which have the same,m + 1, 2m ≤n, first leading coefficients. Form=1, we obtain the polynomials discovered by Freund and Ruschewyh. Furthermore, corresponding results are obtained with respect to weight functions of the type 1/√ρl, whereρl is a polynomial positive on [−1, 1].     

Remez-, Nikolskii-, and Markov-type inequalities for generalized nonnegative polynomials with restricted zeros

Sharp Remez-, Nikolskii-, and Markov-type inequalities are proved for functions of the form

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.     

Direct and inverse theorems for Bernstein polynomials in the space of riemann integrable functions

Equivalence theorems concerning the convergence of the Bernstein polynomialsB _n f are well known for continuous functionsf in the sup-norm. The purpose of this paper is to extend these results for functionsf, Riemann integrable on [0, 1], We have therefore to consider the seminorm

Smooth interpolation of a mesh of curves

The interpolation of a mesh of curves by a smooth regularly parametrized surface with one polynomial piece per facet is studied. Not every mesh with a well-defined tangent plane at the mesh points has such an interpolant: the curvature of mesh curves emanating from mesh points with an even number of neighbors must satisfy an additional vertex enclosure constraint. The constraint is weaker than previous analyses in the literature suggest and thus leads to more efficient constructions. This is illustrated by an implemented algorithm for the local interpolation of a cubic curve mesh by a piecewise [bi]quarticC ¹ surface. The scheme is based on an alternative sufficient constraint that forces the mesh curves to interpolate second-order data at the mesh points. Rational patches, singular parametrizations, and the splitting of patches are interpreted as techniques to enforce the vertex enclosure constraint.Communicated by Wolfgang Dahmen.     

Christoffel functions,orthogonal polynomials,and Nevai's conjecture for Freud weights

We obtain upper and lower bounds for Christoffel functions for Freud weights by relatively new methods, including a new way to estimate discretization of potentials. We then deduce bounds for orthogonal polynomials on thereby largely resolving a 1976 conjecture of P. Nevai. For example, let W:=e ^–Q, whereQ: is even and continuous in, Q^" is continuous in (0, ) andQ ^'>0 in (0, ), while, for someA, B,

Spline interpolation at knot averages

It is well known that when interpolation points coincide with knots, the knot sequence must obey some restriction in order to guarantee the existence and boundedness of the interpolation projector. But, when the interpolation points are chosen to be the knot averages, the corresponding quadratic or cubic spline interpolation projectors are bounded independently of the knot sequence. Based on this fact, de Boor in 1975 made a conjecture that interpolation by splines of orderk at knot averages is bounded for anyk. In this paper we disprove de Boor's conjecture fork 20.Communicated by Wolfgang Dahmen.     

On the sensitivity of radial basis interpolation to minimal data separation distance

Motivated by the problem of multivariate scattered data interpolation, much interest has centered on interpolation by functions of the form

Erratum: Christoffel functions,orthogonal polynomials,and Nevai's conjecture for Freud weights

Communicated by Edward B. Saff     

Norm behavior and zero distribution for orthogonal polynomials with nonsymmetric weights

The asymptotic behavior of thenth root of the leading coefficient of orthogonal polynomials on (–,) and the distribution of their zeros is studied for nonsymmetric weights that behave like exp(–2B|x|) whenx>0 and exp(–2Ax ) whenx<0,>AB. These results generalize previous investigations of Rakhmanov and Mhaskar and Saff who handle the symmetric caseA=B.Communicated by Paul Nevai.     

Chebyshev polynomials for disjoint compact sets

We are concerned with the problem of finding among all polynomials of degreen with leading coefficient 1, the one which has minimal uniform norm on the union of two disjoint compact sets in the complex plane. Our main object here is to present a class of disjoint sets where the best approximation can be determined explicitly for alln. A closely related approximation problem is obtained by considering all polynomials that have degree no larger thann and satisfy an interpolatory constraint. Such problems arise in certain iterative matrix computations. Again, we discuss a class of disjoint compact sets where the optimal polynomial is explicitly known for alln.Communicated by Doron S. Lubinsky     

Closed surfaces defined from biquadratic splines

In order to construct closed surfaces with continuous unit normal, this paper studies certain spaces of spline functions on meshes of four-sided faces. The functions restricted to the faces are biquadratic polynomials or, in certain special cases, bicubic polynomials. A basis is constructed of positive functions with small support which sum to 1 and reduce to tensor-product biquadratic B-splines away from certain singular vertices. It is also shown that the space is suitable for interpolating data at the midpoints of the faces.Communicated by Wolfgang Dahmen.     

On the structure of a table of multivariate rational interpolants

In the table of multivariate rational interpolants the entries are arranged such that the row index indicates the number of numerator coefficients and the column index the number of denominator coefficients. If the homogeneous system of linear equations defining the denominator coefficients has maximal rank, then the rational interpolant can be represented as a quotient of determinants. If this system has a rank deficiency, then we identify the rational interpolant with another element from the table using less interpolation conditions for its computation and we describe the effect this dependence of interpolation conditions has on the structure of the table of multivariate rational interpolants. In the univariate case the table of solutions to the rational interpolation problem is composed of triangles of so-called minimal solutions, having minimal degree in numerator and denominator and using a minimal number of interpolation conditions to determine the solution.Communicated by Dietrich Braess.     

A necessary and sufficient condition for the linear independence of the integer translates of a compactly supported distribution

Given a multivariate compactly supported distribution, we derive here a necessary and sufficient condition for the global linear independence of its integer translates. This condition is based on the location of the zeros of =the Fourier-Laplace transform of. The utility of the condition is demonstrated by several examples and applications, showing, in particular, that previous results on box splines and exponential box splines can be derived from this condition by a simple combinatorial argument.Communicated by Carl de Boor.     

Some properties of A-spaces and their relationship toL 1-approximation

Two new characterizations of A-spaces on an interval are obtained establishing a connection between the A-property and the Hobby-Rice theorem. A complete characterization of tensor product A-spaces on a rectangle is also given.Communicated by Allan Pinkus.     

On the L 1-condition number of the univariate Bernstein basis

We show that the size of the 1-norm condition number of the univariate Bernstein basis for polynomials of degree n is O (2ⁿ / √n). This is consistent with known estimates [3], [5] for p = 2 and p = ∞ and leads to asymptotically correct results for the p-norm condition number of the Bernstein basis for any p with 1 ≤ p ≤ ∞.