Interpolatory Pointwise Estimates for Polynomial Approximation

We discuss whether or not it is possible to have interpolatory pointwise estimates in the approximation of a function f∈ C[0,1] , by polynomials. For the sake of completeness, as well as in order to strengthen some existing results, we discuss briefly the situation in unconstrained approximation. Then we deal with positive and monotone constraints where we show exactly when such interpolatory estimates are achievable by proving affirmative results and by providing the necessary counterexamples in all other cases. November 16, 1998. Date revised: July 12, 1999. Date accepted: September 13, 1999.     

More on Comonotone Polynomial Approximation

The main achievement of this paper is that we show, what was to us, a surprising conclusion, namely, twice continuously differentiable functions in (0,1) (with some regular behavior at the endpoints) which change monotonicity at least once in the interval, are approximable better by comonotone polynomials, than are such functions that are merely monotone. We obtain Jackson-type estimates for the comonotone polynomial approximation of such functions that are impossible to achieve for monotone approximation. July 7, 1998. Date revised: May 5, 1999. Date accepted: July 23, 1999.     

Note on the Approximation of Powers of the Distance in Two-Dimensional Domains

Although Newman's trick has been mainly applied to the approximation of univariate functions, it is also appropriate for the approximation of multivariate functions that are encountered in connection with Green's functions for elliptic differential equations. The asymptotics of the real-valued function on a ball in 2-space coincides with that for an approximation problem in the complex plane. The note contains an open problem. May 17, 1999. Date revised: October 20, 1999. Date accepted: March 17, 2000.     

Polynomial Approximation on Convex Subsets of R n

Let K be a closed bounded convex subset of R ⁿ ; then by a result of the first author, which extends a classical theorem of Whitney there is a constant w _m (K) so that for every continuous function f on K there is a polynomial ϕ of degree at most m-1 so that |f(x)-ϕ(x)|≤ w_m(K) sup _{x,x+mh∈ K} |Δ_h^m(f;x)|. The aim of this paper is to study the constant w _m (K) in terms of the dimension n and the geometry of K . For example, we show that w ₂ (K)≤ (1/2) [ log ₂ n]+5/4 and that for suitable K this bound is almost attained. We place special emphasis on the case when K is symmetric and so can be identified as the unit ball of finite-dimensional Banach space; then there are connections between the behavior of w _m (K) and the geometry (particularly the Rademacher type) of the underlying Banach space. It is shown, for example, that if K is an ellipsoid then w ₂ (K) is bounded, independent of dimension, and w ₃ (K)\sim log n . We also give estimates for w ₂ and w ₃ for the unit ball of the spaces l _p ⁿ where 1≤ p≤∈fty. September 24, 1997. Dates revised: January 18, 1999 and June 10, 1999. Date accepted: June 25, 1999.     

Restricted Nonlinear Approximation

We introduce a new form of nonlinear approximation called restricted approximation . It is a generalization of n -term wavelet approximation in which a weight function is used to control the terms in the wavelet expansion of the approximant. This form of approximation occurs in statistical estimation and in the characterization of interpolation spaces for certain pairs of L _p and Besov spaces. We characterize, both in terms of their wavelet coefficients and also in terms of their smoothness, the functions which are approximated with a specified rate by restricted approximation. We also show the relation of this form of approximation with certain types of thresholding of wavelet coefficients. March 31, 1998. Date accepted: January 28, 1999.     

Weighted Polynomial Approximation for Convex External Fields

It is proven that if Q is convex and w(x)= exp(-Q(x)) is the corresponding weight, then every continuous function that vanishes outside the support of the extremal measure associated with w can be uniformly approximated by weighted polynomials of the form w ⁿ P _n . This solves a problem of P. Borwein and E. B. Saff. Actually, a similar result is true locally for any parts of the extremal support where Q is convex. February 10, 1998. Date revised: July 23, 1998. Date accepted: August 17, 1998.     

Adaptive Mollifiers for High Resolution Recovery of Piecewise Smooth Data from its Spectral Information

We discuss the reconstruction of piecewise smooth data from its (pseudo-) spectral information. Spectral projections enjoy superior resolution provided the data is globally smooth, while the presence of jump discontinuities is responsible for spurious O (1) Gibbs oscillations in the neighborhood of edges and an overall deterioration of the unacceptable first-order convergence in rate. The purpose is to regain the superior accuracy in the piecewise smooth case, and this is achieved by mollification. Here we utilize a modified version of the two-parameter family of spectral mollifiers introduced by Gottlieb and Tadmor [GoTa85]. The ubiquitous one-parameter, finite-order mollifiers are based on dilation . In contrast, our mollifiers achieve their high resolution by an intricate process of high-order cancellation . To this end, we first implement a localization step using an edge detection procedure [GeTa00a, b]. The accurate recovery of piecewise smooth data is then carried out in the direction of smoothness away from the edges, and adaptivity is responsible for the high resolution. The resulting adaptive mollifier greatly accelerates the convergence rate, recovering piecewise analytic data within exponential accuracy while removing the spurious oscillations that remained in [GoTa85]. Thus, these adaptive mollifiers offer a robust, general-purpose ``black box' procedure for accurate post-processing of piecewise smooth data. March 29, 2001. Final version received: August 31, 2001.     

Approximation by boolean sums of positive linear operators III: Estimates for some numerical approximation schemes

In the present note we intröduce and investigate certain sequences of discrete positive linear operators and Boolean sum modifications of them. The mappings considered are obtained by discretizing a class of transformed convolution-type operators using Gaussian quadrature of appropriate order. For our operators and their modifications we prove pointwise Jackson-type theorems involving the first and second order moduli of smoothness, thus providing new and elegant proofs of earlier results by Timan, Telyakowskii, Gopengauz and DeVore. Due to their discrete structure, optimal order of approximation and ease of computation, the operators appear to be useful for numerical approximation. In an intermediate step we solve an old problem in Approximation Theory; its importance was only recently emphasized in a paper of Butzer.     

Constrained Approximation in Banach Spaces

   Abstract. We consider the problem of approximating vectors from a complemented subspace Z ₊ of a Banach space X by the projections onto Z ₊ of vectors from a subspace Y ₊ with a norm constraint on their projections onto the complementary subspace. Sufficient conditions are found for the existence of a unique best approximant and a characterization via a critical point equation is provided, thus extending known results on Hilbert spaces. These results are then applied in the case that X is L ^p (T), where T denotes the unit circle, Z ₊ consists of functions supported on a subset of the circle, and Y ₊ is the corresponding Hardy space.     

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.     

On Multivariate Adaptive Approximation

Recently, A. Cohen, R. A. DeVore, P. Petrushev, and H. Xu investigated nonlinear approximation in the space BV (R ² ). They modified the classical adaptive algorithm to solve related extremal problems. In this paper, we further study the modified adaptive approximation and obtain results on some extremal problems related to the spaces V _σ,p ^r (R ^d ) of functions of ``Bounded Variation" and Besov spaces B ^α (R ^d ). November 23, 1998. Date revised: June 25, 1999. Date accepted: September 13, 1999.     

Shape-Preserving Widths of Weighted Sobolev-Type Classes of Positive, Monotone, and Convex Functions on a Finite Interval

   Abstract. Let I be a finite interval, r∈ N and ρ(t)= dist {t, I} , t∈ I . Denote by Δ ^s ₊ L _q the subset of all functions y∈ L _q such that the s -difference Δ ^s _τ y(t) is nonnegative on I ,

Rates of Best Uniform Rational Approximation of Analytic Functions by Ray Sequences of Rational Functions

In this paper, problems related to the approximation of a holomorphic function f on a compact subset E of the complex plane C by rational functions from the class of all rational functions of order (n,m) are considered. Let ρ _n,m = ρ _n,m (f;E) be the distance of f in the uniform metric on E from the class . We obtain results characterizing the rate of convergence to zero of the sequence of the best rational approximation { ρ _n,m(n) } _n=0 ^∞ , m(n)/n → θ ∈ (0,1] as n → ∞ . In particular, we give an upper estimate for the liminf _{n →∞} ρ _n,m(n) ^1/(n+m(n)) in terms of the solution to a certain minimum energy problem with respect to the logarithmic potential. The proofs of the results obtained are based on the methods of the theory of Hankel operators. June 16, 1997. Date revised: December 1, 1997. Date accepted: December 1, 1997. Communicated by Ronald A. DeVore.     

On Stable Local Bases for Bivariate Polynomial Spline Spaces

Stable locally supported bases are constructed for the spaces \cal S _d ^r (\triangle) of polynomial splines of degree d≥ 3r+2 and smoothness r defined on triangulations \triangle , as well as for various superspline subspaces. In addition, we show that for r≥ 1 , in general, it is impossible to construct bases which are simultaneously stable and locally linearly independent. February 2, 2000. Date revised: November 27, 2000. Date accepted: March 7, 2001.     

On the Best Approximation by Ridge Functions in the Uniform Norm

We consider the best approximation of some function classes by the manifold M _n consisting of sums of n arbitrary ridge functions. It is proved that the deviation of the Sobolev class W _p ^r,d from the manifold M _n in the space L _q for any 2≤ q≤ p≤∈fty behaves asymptotically as n ^-r/(d-1) . In particular, we obtain this asymptotic estimate for the uniform norm p=q=∈fty . January 10, 2000. Date revised: March 1, 2001. Date accepted: March 12, 2001.     

Best Meromorphic Approximation of Markov Functions on the Unit Circle

Let E subset(-1,1) be a compact set, let μ be a positive Borel measure with support supp μ =E , and let H _p (G), 1≤ p ≤∈fty, be the Hardy space of analytic functions on the open unit disk G with circumference Γ={z colon |z|=1} . Let Δ _n,p be the error in best approximation of the Markov function frac{1}{2π i} ∈t_E frac{d μ(x)}{z-x} in the space L _p (Γ) by meromorphic functions that can be represented in the form h=P/Q , where P ∈ H _p (G), Q is a polynomial of degree at most n , Qnot equiv 0 . We investigate the rate of decrease of Δ _n,p , 1≤ p ≤∈fty , and its connection with n -widths. The convergence of the best meromorphic approximants and the limiting distribution of poles of the best approximants are described in the case when 1<p≤∈fty and the measure μ with support E=[a,b] satisfies the Szegő condition ∈t_a^b frac{log(d μ/ d x)}{sqrt{(x-a)(b-x)}} dx >- ∈fty. July 27, 2000. Final version received: May 19, 2001.     

On the Power of Standard Information for Weighted Approximation

We study weighted approximation of multivariate functions for classes of standard and linear information in the worst case and average case settings. Under natural assumptions, we show a relation between n th minimal errors for these two classes of information. This relation enables us to infer convergence and error bounds for standard information, as well as the equivalence of tractability and strong tractability for the two classes. April 11, 2001. Final version received: May 29, 2001.     

A Bound on the Approximation Order of Surface Splines

The functions φ _m :=|.| ^2m-d if d is odd, and φ _m :=|.| ^2m-d \log|.| if d is even, are known as surface splines, and are commonly used in the interpolation or approximation of smooth functions. We show that if one's domain is the unit ball in R ^d , then the approximation order of the translates of φ _m is at most m . This is in contrast to the case when the domain is all of R ^d where it is known that the approximation order is exactly 2m . April 23, 1996. Date revised: May 5, 1997.     

On Approximation by Ridge Functions

Ridge functions are defined as functions of the form , where , belongs to the given ``direction' set . In this paper we study the fundamentality of ridge functions for variable directions sets A and discuss the rate of approximation by ridge functions. Date received: June 7, 1994. Date revised: August 3, 1995.     

Compactly Supported, Piecewise Affine Scaling Functions on Triangulations

The methods of using intertwining MRAs to find orthogonal scaling functions have previously been applied to one-dimensional MRAs and are here extended to two-dimensional bases. Two examples are constructed from MRAs consisting of continuous, compactly supported, piecewise affine functions of two variables. The resulting scaling functions can be conveniently restricted to compact domains. November 23, 1997. Date accepted: January 22, 1999.