On the uniqueness of canonical points in the Hobby-Rice theorem

According to the Hobby-Rice theorem for anyn-dimensional subspaceU _n ofL ₁([a, b], ) ( positive, finite, nonatomic) there exist points =s ₀x ₁x _m+1=b, where 0mn, such that

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 ≤ ∞.     

Ratio asymptotics and quadrature formulas

A technique to find the asymptotic behavior of the ratio between a polynomialss _n and thenth orthonormal polynomial with respect to a positive measureμ is shown. Using it, some new results are found and a very simple proof for other classics is given.     

On the asymptotic distribution of eigenvalues of banded matrices

We consider the abstract measures, known as thedensity- of- states measures, associated with the asymptotic distribution of eigenvalues of infinite banded Hermitian matrices. Two widely used definitions of these measures are shown to be equivalent, even in the unbounded case, and we prove that the density of states is invariant under certain, possibly unbounded, perturbations. Also considered are measures associated with the asymptotic distribution of eigenvalues of rescaled unbounded matrices. These measures are associated with the so-called contracted spectrum when the matrices are tridiagonal. Finally, we produce several examples clarifying the nature of the density of states.Communicated by Paul Nevai.     

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.     

On a discrete norm of the self-inversive unimodular polynomials

The quotient of divided by , whereP is a self-inversive and unimodular polynomial of any degree, dominates an absolute constantK>1. A 1989 paper gaveK=1.0252 on which its authors conjetured that the best constant is . We supply counter examples to their claim and provide a partial result for whenever theL _q norm is replaced by some “discrete” type norm. Research supported by the Shiraz University Grant 72-SC-784-432.     

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

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.     

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.     

The Gibbs phenomenon for bestL 1-trigonometric polynomial approximation

The classical Gibbs phenomenon for the Fourier sections (bestL ₂-trigonometric polynomial approximants) of a jump function asserts that, near the jump, these sections overshoot the function by an asymptotically constant factorg (theL ₂-Gibbs constant). In this paper we show that, for a class of one-jump discontinuous functions, a similar phenomenon holds for the trigonometric polynomials of bestL ₁-approximation. We determine theL ₁-Gibbs constant , which is substantially smaller thang. Furthermore, we prove that uniform convergence of bestL ₁-approximants takes place on intervals that avoid the jump. In the analysis we obtain some strong uniqueness theorems for bestL ₁-approximants.Communicated by Vladimir N. Temlyakov.     

Best polynomial approximation in Sobolev-Laguerre and Sobolev-Legendre spaces

We investigate limiting behavior as γ tends to ∞ of the best polynomial approximations in the Sobolev-Laguerre space W^N,2([0, ∞); e^−x) and the Sobolev-Legendre space W^N,2([−1, 1]) with respect to the Sobolev-Laguerre inner product

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

We prove that an absolute constantc>0 exists such that

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.     

Spaces generated by smooth blocks

A new kind of function space generated by smooth blocks is introduced. The new spaces are used to investigate the relation between the smoothness imposed on blocks and the rate of convergence of the Bochner-Riesz means at the critical index.Communicated by Robert C. Sharpley.     

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.     

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.     

Gibbs-Wilbraham splines

If a function with a jump discontinuity is approximated in the norm ofL ²[–1,1] by a periodic spline of orderk with equidistant knots, a behavior analogous to the Gibbs-Wilbraham phenomenon for Fourier series occurs. A set of cardinal splines which play the role of the sine integral function of the classical phenomenon is introduced. It is then shown that ask becomes large, the phenomenon for splines approaches the classical phenomenon.Communicated by Ronald A. DeVore.     

Adaptive greedy approximations

The problem of optimally approximating a function with a linear expansion over a redundant dictionary of waveforms is NP-hard. The greedy matching pursuit algorithm and its orthogonalized variant produce suboptimal function expansions by iteratively choosing dictionary waveforms that best match the function’s structures. A matching pursuit provides a means of quickly computing compact, adaptive function approximations. Numerical experiments show that the approximation errors from matching pursuits initially decrease rapidly, but the asymptotic decay rate of the errors is slow. We explain this behavior by showing that matching pursuits are chaotic, ergodic maps. The statistical properties of the approximation errors of a pursuit can be obtained from the invariant measure of the pursuit. We characterize these measures using group symmetries of dictionaries and by constructing a stochastic differential equation model. We derive a notion of the coherence of a signal with respect to a dictionary from our characterization of the approximation errors of a pursuit. The dictionary elements slected during the initial iterations of a pursuit correspond to a function’s coherent structures. The tail of the expansion, on the other hand, corresponds to a noise which is characterized by the invariant measure of the pursuit map. When using a suitable dictionary, the expansion of a function into its coherent structures yields a compact approximation. We demonstrate a denoising algorithm based on coherent function expansions.     

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.     

On maximal ranges of polynomial spaces in the unit disk

Let be a domain in C, 0, and let _n ⁰ () be the set of polynomials of degreen such thatP(0)=0 andP(D), whereD denotes the unit disk. The maximal range _n is then defined to be the union of all setsP(D),P _n ⁰ (). We derive necessary and, in the case of ft convex, sufficient conditions for extremal polynomials, namely those boundaries whose ranges meet _n . As an application we solve explicitly the cases where is a half-plane or a strip-domain. This also implies a number of new inequalities, for instance, for polynomials with positive real part inD. All essential extremal polynomials found so far in the convex cases are univalent inD. This leads to the formulation of a problem. It should be mentioned that the general theory developed in this paper also works for other than polynomial spaces.Communicated by J. Milne Anderson.