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

Optimality and uniqueness conditions in complex rational Chebyshev approximation with examples

It is well known that best complex rational Chebyshev approximants are not always unique and that, in general, they cannot be characterized by the necessary local Kolmogorov condition or by the sufficient global Kolmogorov condition. Recently, Ruttan (1985) proposed an interesting sufficient optimality criterion in terms of positive semidefiniteness of some Hermitian matrix. Moreover, he asserted that this condition is also necessary, and thus provides a characterization of best approximants, in a fundamental case.In this paper we complement Ruttan's sufficient optimality criterion by a uniqueness condition and we present a simple procedure for computing the set of best approximants in case of nonuniqueness. Then, by exhibiting an approximation problem on the unit disk, we point out that Ruttan's characterization in the fundamental case is not generally true. Finally, we produce several examples of best approximants on a real interval and on the unit circle which, among other things, give some answers to open questions raised in the literature.     

Rational approximation to Lipschitz and Zygmund classes

In this paper, characterizations for lim _n(R _n (f)/(n ^–1)=0 inH and for lim _n(n ^r+ R _n (f)=0 inW ^r Lip ,r1, are given, while, forZ, a generalization to a related result of Newman is established.Communicated by Ronald A. DeVore.     

Incomplete rational approximation in the complex plane

We consider rational approximations of the form

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.     

Absolute projection constant of the linear functions in a Lebesgue space

Cheney and Franchetti have computed the projection constant (V; L ¹[a, b]) for the two-dimensional spaceV of lines inL ¹[a, b] relative to the overspaceL ¹[a, b]. The purpose of this note is to show that (V; L ¹[a, b]) is in fact the absolute projection constant ofV.Communicated by Carl de Boor.     

Distribution of poles of diagonal rational approximants to functions of fast rational approximability

We show that the most of the time, most poles of diagonal multipoint Padé or best rational approximants to functions admitting fast rational approximation, leave the region of meromorphy. Following is a typical result: Letf be single-valued and analytic in CS, where cap(S)=0. Let {n _j } _j=1 be an increasing sequence of positive integers withn _j+1/n _j 1 asj. Then there exists an infinite sequenceL of positive integers such that asj,jL the total multiplicity of poles of any sequence of type (n _j ,n _j ) multipoint Padé or best rational approximants tof, iso(n _j ) in any compactK in whichf is meromorphic. The sequenceL is independent of the particular sequence of multipoint Padé or best approximants, and yields the same behavior for near-best approximants. If the errors of best approximation on some compact set satisfy a weak regularity condition, then we may takeL={1,2,3,}.Communicated by Edward B. Saff.     

Weighted polynomial approximation with Freud weights

We show that ifw(x)=exp(–|x|), then in the case =1 for every continuousf that vanishes outside the support of the corresponding extremal measure there are polynomialsP _n of degree at mostn such thatw ⁿ P _n uniformly tends tof, and this is not true when <1. these=" are=" the=" missing=" cases=" concerning=" approximation=" by=" weighted=" polynomials=" of=" the=">w ⁿ P _n wherew is a Freud weight. Our second theorem shows that even if we are only interested in approximation off on the extremal support, the functionf must still vanish at the endpoints, and we actually determine the (sequence of) largest possible intervals where approximation is possible. We also briefly discuss approximation by weighted polynomials of the formW(a_nx)P _n (x).Communicated by Edward B. Saff.     

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.     

Interpolatory properties of best rationalL 1-approximations

In this paper the distribution of the zeros of the error function for bestL ₁-approximation by rational functions fromR _n,m is considered. It is shown that the maximal distance between such zeros isO(1/(n–m)), ifn > m.Communicated by Edward B. Saff.     

On estimates of approximation numbers and best bilinear approximation

We obtain estimates of approximation numbers of integral operators, with the kernels belonging to Sobolev classes or classes of functions with bounded mixed derivatives. Along with the estimates of approximation numbers, we also obtain estimates of best bilinear approximation of such kernels.Communicated by Charles A. Micchelli.     

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

We prove that an absolute constantc>0 exists such that

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

Whitney theorem of interpolatory type for k-monotone functions

One of the main results of this paper is the following Whitney theorem of interpolatory type for k-monotone functions (i.e., functions f such that divided differences f[x ₀,…, x _k ] are nonnegative for all choices of (k + 1) distinct points x ₀,…, x _k .     

Optimal ray sequences of rational functions connected with the Zolotarev problem

Given two compact disjoint subsetsE ₁,E ₂ of the complex plane, the third problem of Zolotarev concerns estimates for the ratio

An upper bound on the approximation power of principal shift-invariant spaces

An upper bound on theL _p-approximation power (1 ≤p ≤ ∞) provided by principal shift-invariant spaces is derived with only very mild assumptions on the generator. It applies to both stationary and nonstationary ladders, and is shown to apply to spaces generated by (exponential) box splines, polyharmonic splines, multiquadrics, and Gauss kernel.     

Best uniform approximation by harmonic functions on subsets of Riemannian manifolds

We investigate best uniform approximations to bounded, continuous functions by harmonic functions on precompact subsets of Riemannian manifolds. Applications to approximation on unbounded subsets ofR ² are given.Communicated by J. Milne Anderson.     

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 ≤ ∞.     

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.