Problems In Approximation Theory Research Articles

Sparse representations and approximation theory

This paper is an attempt to both expound and expand upon, from an approximation theorist’s point of view, some of the theoretical results that have been obtained in the sparse representation (compressed sensing) literature. In particular, we consider in detail ℓ 1 m -approximation, which is fundamental in the theory of sparse representations, and the connection between the theory of sparse representations and certain n -width concepts. We try to illustrate how the theory of sparse representation leads to new and interesting problems in approximation theory, while the results and techniques of approximation theory can further add to the theory of sparse representations.

Properties of reciprocal derivatives

The continued fraction expansion of functions is one of important problems in approximation theory along with approximations by power series, orthogonal polynomials, Pade approximations, etc., because expansions of this type can be used for the calculation of values of functions on a computer [1, 2]. The continued fraction expansion of a function is obtained from the corresponding expansion of this function in a power series by the construction of a regular continued C -fraction corresponding to this power series [3–6]. The continued fraction expansion of a function can also be obtained by using the Thiele formula [7, 8]. The equivalence of these two methods for the expansion of functions in regular continued C -fractions was established in [9]. In 1909, Thiele [7] introduced reciprocal divided differences; using them, he defined reciprocal derivatives of functions of a single real variable and obtained analogs of the Taylor formula in the theory of continued fractions. For properties of reciprocal differences and reciprocal derivatives, see [7, 8, 10]. In the present paper, we establish some new properties of reciprocal derivatives.

A note on the convexity of Chebyshev sets

Perhaps one of the major unsolved problem in Approximation Theory is : Whether or not every Chebyshev subset of a Hilbert space must be convex. Many partial answers to this problem are available in the literature. R.R. Phelps [Proc. Amer. Math. Soc. 8 (1957), 790-797] showed that a Chebyshev set in an inner product space (or in a strictly convex normed linear space) is convex if the associated metric projection is non-expansive. We extend this result to metric spaces.

A probabilistic construction of model validation

We describe a procedure to assess the predictive accuracy of process models subject to approximation error and uncertainty. The proposed approach is a functional analysis-based probabilistic approach for which we represent random quantities using polynomial chaos expansions (PCEs). The approach permits the formulation of the uncertainty assessment in validation, a significant component of the process, as a problem of approximation theory. It has two essential parts. First, a statistical procedure is implemented to calibrate uncertain parameters of the candidate model from experimental or model-based measurements. Such a calibration technique employs PCEs to represent the inherent uncertainty of the model parameters. Based on the asymptotic behavior of the statistical parameter estimator, the associated PCE coefficients are then characterized as independent random quantities to represent epistemic uncertainty due to lack of information. Second, a simple hypothesis test is implemented to explore the validation of the computational model assumed for the physics of the problem. The above validation path is implemented for the case of dynamical system validation challenge exercise.

Problems of approximation theory in linear spaces

We present a survey of results related to the approximation characteristics of the spaces Sϕp and their generalizations. The proposed approach enables one to obtain solutions of problems of classical approximation theory in abstract linear spaces in explicit form. The results obtained yield statements that are new even in the case of approximations in the functional Hilbert spaces L2.

Extremal Problems of Approximation Theory in Linear Spaces

We propose an approach that enables one to pose and completely solve main extremal problems in approximation theory in abstract linear spaces. This approach coincides with the traditional one in the case of approximation of sets of functions defined and square integrable with respect to a given σ-additive measure on manifolds in Rm, m ≥ 1.

Curvelets and Fourier Integral Operators

A recent body of work introduced new tight-frames of curvelets E. Candès, D. Donoho, in: (i) Curvelets – a suprisingly effective nonadaptive representation for objects with edges (A. Cohen, C. Rabut, L. Schumaker (Eds.)), Vanderbilt University Press, Nashville, 2000, pp. 105–120; (ii) http://www.acm.caltech.edu/~emmanuel/publications.html, 2002 to address key problems in approximation theory and image processing. This paper shows that curvelets essentially provide optimally sparse representations of Fourier Integral Operators. To cite this article: E. Candès, L. Demanet, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Cohomology vanishing¶and a problem in approximation theory

For a simplicial subdivisonof a region in k n (k algebraically closed) and r ∈ N, there is a reflexive sheaf K on P n , such that H 0 (K(d)) is essentially the space of piecewise polynomial functions on � , of degree at most d, which meet with order of smoothness r along common faces. In (9), Elencwajg and Forster give bounds for the vanishing of the higher cohomology of a bundle E on P n in terms of the top two Chern classes and the generic splitting type of E. We use a spectral sequence argument similar to that of (16) to characterize thosefor which K is actually a bundle (which is always the case for n = 2). In this situation we can obtain a formula for H 0 (K(d)) which involves only local data; the results of (9) cited earlier allow us to give a bound on the d where the formula applies. We also show that a major open problem in approximation theory may be formulated in terms of a cohomology vanishing on P 2 and we discuss a possible connection between semi-stability and the conjectured answer to this open problem.

On the Best Polynomial Approximations of 2π-Periodic Functions and Exact Values of n-Widths of Functional Classes in the Space L2

To solve extremal problems of approximation theory in the space L2, we use τ-moduli introduced by Ivanov. We determine the exact values of constants in Jackson-type inequalities and the exact values of n-widths of functional classes determined by these moduli.

Solving the discrete lp-approximation problem by a method of centers

The discrete l p -approximation problem is a basic problem in approximation theory and optimization. This problem is normally solved by Newton-type methods which are complicated by the nondifferentiability of the gradient function for p∈[1,2). This paper discusses a scheme and its implementation for solving this problem by a method of analytic centers, which provides a unified treatment for all p⩾1 and has a polynomial bound of complexity. The original problem is reformulated as a constrained convex programming problem which admits a self-concordant logarithmic barrier. A special structure of the Newton system derived from minimizing the potential function is utilized to reduce the amount of computation. Some other implementation techniques are also presented. Computational results show that the method of centers is robust for all p.

JACKSON TYPE THEOREMS FOR COMPACT RANK 1 SYMMETRIC SPACES

Since recently various problems of approximation theory have been actively studied for functions on the n-dimensional sphere Sn (see [1–3] and the bibliography therein). A broader but natural class of spaces for which we can pose such problems is that of all compact rank 1 symmetric spaces (CROSP in the terminology of the monograph [4]). Some results are already available for these spaces (see [5–9]); however, the main problems still remain open. This article is devoted to proving the direct Jackson-type theorems for an arbitrary CROSP M . The complete classification of all CROSPs is well known. It comprises the four series: the spheres Sn (n = 1, 2, . . . ) and the real, complex, and quaternionic projective spaces (Pn(R), n = 2, 3, . . . ; Pn(C), n = 4, 6, . . . ; and Pn(H), n = 8, 12, 16, . . . ) (everywhere the superscript n denotes the dimension of the space) and one special space, the Cayley elliptic plane P 16(Cay). Since the problems of harmonic analysis on Pn(R) are easily reduced to the corresponding problems on Sn, we assume that M 6= Pn(R). A CROSP M is always a Riemannian manifold. Let ∆ be the Laplace–Beltrami operator on M . The spectrum of ∆ is discrete, real, and nonpositive. We arrange it in decreasing order (0 = λ0 > λ1 > λ2 > . . . ) and denote the eigensubspace of ∆ corresponding to the eigenvalue λk by Hk (it is always finite-dimensional). Put PN (M) := H0 + H1 + · · · + HN . The functions in PN (M) are referred to as spherical polynomials on M of degree m (for M = Sn they coincide with the ordinary spherical polynomials). Given a set X with a measure dσ, we as usual denote by Lp(X, dσ) the Banach space of complexvalued measurable functions f(x) on X with the finite norm

New Results on the Equilibrium Measure for Logarithmic Potentials in the Presence of an External Field

In this paper we use techniques from the theory of ODEs and also from inverse scattering theory to obtain a variety of results on the regularity and support properties of the equilibrium measure for logarithmic potentials on the finite interval [−1, 1], in the presence of an external fieldV. In particular, we show that ifVisC2, then the equilibrium measure is absolutely continuous with respect to Lebesgue measure, with a density which is Hölder-12 on (−1, 1), and with at worst a square root singularity at ±1. Moreover, ifVis real analytic then the support of the equilibrium measure consists of a finite number of intervals. In the cases whereV=txm,m=1, 2, 3, or 4, the equilibrium measure is computed explicitly for allt∈R. For these cases the support of the equilibrium measure consists of 1, 2, or 3 intervals, depending ontandm. We also present detailed results for the general monomial caseV=txm, for allm∈N. The regularity results for the equilibrium measure are obtained by careful analysis of the Fekete points associated to the weightenV(x)dx. The results on the support of the equilibrium measure are obtained using two different approaches: (i) an explicit formula of the kind derived by physicists for mean-field theory calculations; (ii) detailed perturbation theoretic results of the kind that are needed to analyze the zero dispersion limit of the Korteweg–de Vries equation in Lax–Levermore theory. The implications of the above results for a variety of related problems in approximation theory and the theory of orthogonal polynomials are also discussed.

Some old and new problems in approximation theory: Research problems 95-1

Some old and new problems in approximation theory: Research problems 95-1

Massiveness of the sets of extremal functions in some problems in approximation theory

It is proved that the sets of extremal functions are massive in some problems in approximation theory.

Approximation by rational modules in Lp and BMO

where each ri(z)~9?(X). Let a= ;(8/2x+ i2/+) be the usual CauchyRiemann operator in the complex plane and let d* = 8 c ii. We note that a function satisfies a*f = 0 in an open set U if and only if f= h + Fk with h and k analytic. A major problem in approximation theory is describing the closure of the rational module 92(X) ~?i in various norms (see [3]). It has been shown in [S] that W(X) gi is dense in C(X), the space of all continuous functions on X, for any compact X when the interior k of X is empty. However, the presence of an interior really complicates the situation. An outstanding open problem is whether S?(X) P, is dense in the space { f~ C(X): J*f= 0 in k} for all compact sets X. For 1 6 p 6 00, let L,P(X) be the closed subspace of Lp(X) which consists of functions analytic in 2. If I/ is any space of functions on X, we denote by [ VJP the closure of V with respect to the LP(X) norm, and we write p = {f: f~ V} where p is the usual Cauchy transform of ,fi In (6) the author proved that [9(.X’) g,], = [W(X) + L:(X)” 1, for all 1 d p < K. using the theory of singular integrals and Schwartz lemma. In this note, we use a somewhat constructuve method to prove the same result for the Lp norms and then extend it to the BMO norm case. To each compact subset X of @ (E lR*) one associates the restrictive spaces BMO(X) = BMO(C)lX z BMO(C))lI(X) and VMO(X) = VMO(@)~,EVMO(C)/J(X), where Z(X)= {~EBMO(C): f’=O a.e. on Xi and J(X)= {~EVMO(@): f=O a.e. on X}. Let [9(X) P,],,, and [g(x) + L,” cm A 1 BMCI be the closure in BMO(X) of 9(X) 9, and a(X) + L,:(X) h, respectively. Holden’s extension theorem [ 1 ] implies that

OSCILLATION PROPERTIES OF DIFFERENTIAL OPERATORS AND CONVOLUTION OPERATORS, AND SOME APPLICATIONS

A criterion is obtained for the validity of Rolle's theorem on the interval for linear differential operators. The exact order is found of decrease of the Fourier coefficients of the convolution kernels guaranteeing the variation diminishing property. Applications are given to several problems of approximation theory (such as quadrature formulas, recovering, widths, trigonometric interpolation, etc.). Bibliography: 24 titles.

Weighted polynomials, radial basis functions and potentials on locally compact spaces

We discuss the classical notions of capacity, transfinite diameter, Chebyshev constant and equilibrium distributions in the context of locally compact spaces. We also demonstrate how the analogues of certain theorems in classical potential theory can be extended to this framework and applied to various problems in approximation theory. In particular, we explore the similarities between the theory of weighted polynomials and the theory of radial basis functions from a potential theoretic perspective.

A variational description of the spectra of totally positive matrices, and extremal problems of approximation theory

A variational description of the spectra of totally positive matrices, and extremal problems of approximation theory

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

In the present note we introduce 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.

On solutions of some nonlinear recurrences

We investigate when non-linear recurrences in vector lattices (that is, partially ordered sets which are vector spaces, and in which every two elements have a maximum and a minimum) have unique positive solutions. As an application of these techniques we study positive solutions of real recurrences which arise in some problems in approximation theory. A typical recurrence of this type is n a n = a n + 1 + a n + a n − 1 for n ⩾ 1 with a 0 = 0 , and the question is whether there is a solution with a n > 0 for all n ⩾ 1.