Showing papers in "Constructive Approximation in 1997"

Adaptive greedy approximations

Abstract: 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.

Approximation order provided by refinable function vectors

Abstract: In this paper we considerLp-approximation by integer translates of a finite set of functionsϕv (v=0, ...,r − 1) which are not necessarily compactly supported, but have a suitable decay rate. Assuming that the function vectorϕ=(ϕ=0/r−1 is refinable, necessary and sufficient conditions for the refinement mask are derived. In particular, if algebraic polynomials can be exactly reproduced by integer translates ofϕv, then a factorization of the refinement mask ofϕ can be given. This result is a natural generalization of the result for a single functionϕ, where the refinement mask ofϕ contains the factor ((1 +e−iu)/2)m if approximation orderm is achieved.

Biorthogonal Wavelet Expansions

Abstract: This paper is concerned with developing conditions on a given finite collection of compactly supported algebraically linearly independent refinable functions that insure the existence of biorthogonal systems of refinable functions with similar properties. In particular, we address the close connection of this issue with stationary subdivision schemes.

Rates of convex approximation in non-hilbert spaces

Abstract: This paper deals with sparse approximations by means of convex combinations of elements from a predetermined “basis” subsetS of a function space. Specifically, the focus is on therate at which the lowest achievable error can be reduced as larger subsets ofS are allowed when constructing an approximant. The new results extend those given for Hilbert spaces by Jones and Barron, including, in particular, a computationally attractive incremental approximation scheme. Bounds are derived for broad classes of Banach spaces; in particular, forL p spaces with 1

Product formulas and associated hypergroups for orthogonal polynomials on the simplex and on a parabolic biangle

Abstract: Explicit product formulas are obtained for families of multivariate polynomials which are orthogonal on simplices and on a parabolic biangle in ${\Bbb R}^2$ . These product formulas are shown to give rise to measure algebras which are hypergroups. The article also includes an elementary proof that the Michael topology for the space of compact subsets of a topological space (which is used in the definition of a hypergroup) is equivalent to the Hausdorff metric topology when the underlying space has a metric.

Jackson and smoothness theorems for freud weights

Abstract: We prove Jackson, realization, and converse theorems for Freud weights inL p, 0

The Diagonal of the Padé Table and the Approximation of the Weyl Function of Second-Order Difference Operators

Abstract: We study connections between diagonal Pade approximants and spectral properties of second-order difference operators with complex coefficients. In the first part, we identify the diagonal of the Pade table with a particular difference operator which is shown to have a maximal resolvent set. The spectrum of an asymptotically periodic complex difference operator is given, and we prove convergence on the resolvent set for the corresponding sequence of Pade approximants to the associated Weyl function.

On Approximation by Ridge Functions

Abstract: Ridge functions are defined as functions of the form \(f(\mbox{\footnotesize\bf a}\cdot \mbox{\footnotesize\bf x})\) , where \(f\colon\ {\Bbb R}\rightarrow {\Bbb R}\) , \(\mbox{\footnotesize\bf x}\in {\Bbb R}^k$, and $\mbox{\footnotesize\bf a}\) belongs to the given ``direction'' set \(A\subset {\Bbb R}^k\) . In this paper we study the fundamentality of ridge functions for variable directions sets A and discuss the rate of approximation by ridge functions.

Spectral Measures and Jacobi Matrices Related to Laguerre-Type Systems of Orthogonal Polynomials

Abstract: This paper considers systems of Laguerre-type orthogonal polynomials for which the corresponding Jacobi matrices represent unbounded self-adjoint operators which are bounded above or below. Under appropriate assumptions on the coefficient sequences in the recursion formula, results are obtained on the uniform boundedness of the polynomials on bounded intervals, the absence of eigenvalues for the corresponding operator, and the absolute continuity of the measure of orthogonality.

Conjectures around the Baker-Gammel-Wills Conjecture

Abstract: The Baker-Gammel-Wills Conjecture states that if a functionf is meromorphic in a unit diskD, then there should, at least, exist an infinite subsequenceN ⊆N such that the subsequence of diagonal Pade approximants tof developed at the origin with degrees contained inN converges tof locally uniformly inD/{poles off} Despite the fact that this conjecture may well be false in the general Pade approximation in several respects In the present paper, six new conjectures about the convergence of diagonal Pade approximants are formulated that lead in the same direction as the Baker-Gammel-Wills Conjecture However, they are more specific and they are based on partial results and theoretical considerations that make it rather probable that these new conjectures hold true

A Discrepancy Theorem on Quasiconformal Curves

Abstract: In [7], Blatt and Mhaskar estimated the Erdős-Turan type discrepancy of a signed Borel measure σ on a sufficiently smooth Jordan curve or arc L in terms of the logarithmic potential of σ on a curve enclosing L. We extend this result to a measure σ on an arbitrary quasiconformal curve. As applications, estimates for the distribution of simple zeros of monic polynomials, Fekete points, extreme points of polynomials of best uniform approximation are obtained.

Fundamental sets of continuous functions on spheres

Abstract: LetSm andS∞ denote the unit spheres inRm+1 ande2, respectively. We look for functionsf inC[−1, 1] such that the family of functionsx →f( ) is fundamental in the spaceC(Sm). Herev runs overSm. There is a similar question forC(S∞), when this space is given the topology of uniform convergence on compact sets.

On Rational Interpolation to |x|

Abstract: We consider Newman-type rational interpolation to |x| induced by arbitrary sets of interpolation nodes, and we show that under mild restrictions on the location of the interpolation nodes, the corresponding sequence of rational interpolants converges to |x|.

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

Abstract: An upper bound on theLp-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

Kergin Interpolants of Holomorphic Functions

Abstract: Let D be a C-convex domain in Cn. Let \(\{A_{dj}\}, \ j = 0,\ldots,d\) , and d = 0,1,2, ..., be an array of points in a compact set \(K \subset D\) . Let f be holomorphic on \(\overline D\) and let Kd(f) denote the Kergin interpolating polynomial to f at Ad0,... , Add. We give conditions on the array and D such that \(\lim_{d\to\infty} \|K_d (f) - f\|_K = 0\) . The conditions are, in an appropriate sense, optimal.

Best Approximation with Walsh Atoms

Abstract: We consider the approximation in L2R of a given function using finite linear combinations of Walsh atoms, which are Walsh functions localized to dyadic intervals, also called Haar—Walsh wavelet packets. It is shown that up to a constant factor, a linear combination of K atoms can be represented to relative error ɛ by a linear combination of \(K^2 \log(1/\varepsilon)\) orthogonal atoms.

Exactn-widths of hardy-sobolev classes

Abstract: Let\(\tilde h^r _{\infty ,\beta } \) and\(\tilde H^r _{\infty ,\beta } \) denote those 2π-periodic, real-valued functions onR that are analytic in the strip |Imz|<β and satisfy the restrictions |Ref(r)(z)| ≤ 1 and |f(r)(z)| ≤ 1, respectively. We determine the Kolmogorov, linear, and Gel’fand widths of\(\tilde h^r _{\infty ,\beta } \) inLq[0, 2π], 1 ≤q ≤ ∞, and\(\tilde H^r _{\infty ,\beta } \) inL∞[0, 2π].

A commuting q-analogue of the addition formula for disk polynomials

Abstract: Starting from the addition formula for q-disk polynomials, which is an identity in noncommuting variables, we establish a basic analogue in commuting variables of the addition and product formula for disk polynomials. These contain, as limiting cases, the addition and product formula for little q-Legendre polynomials. As q tends to 1 the addition and product formula for disk polynomials are recovered.

On unconditional polynomial bases inL p and bergman spaces

Abstract: In this paper we consider unconditional bases inL p(T), 1

L ø-Approximation by nondecreasing functions on the interval

Abstract: Letφ: [0, ∞) → [0, ∞) be a Δ2 convex function,φ(0)=0,φ (y)>0 ify>0, and letf be a Lebesgue measurable function defined on [0, 1], ∫ 0 1 φ(|f|)<∞. We consider the setM o (f) of elements that minimize ∫ 0 1 φ(|f−g|) among the class of nondecreasing functionsg. We give a description of that set, showing that there exists an open setU such that any functiong 0 eM o(f) is constant on each component ofU. Furthermore, whenever the right derivativeφ’+ is bounded, orf is essentially bounded, then anyg 0 coincides withf almost everywhere outsideU, and we show with an example that the former hypotheses are essential for getting that result. We also give a construction of the minimum and maximum elements inM o(f). Finally, we prove that iff is approximately continuous at every point, then there is a uniqueg 0 eM o, andg 0 is continuous. It is also observed that analogous results are valid when approximating with respect to the Luxemburg norm in the associated Orlicz space.

On unconditional polynomial bases in L p and bergman spaces

Abstract: In this paper we consider unconditional bases inL p(T), 1

Ratio asymptotics and quadrature formulas

Abstract: A technique to find the asymptotic behavior of the ratio between a polynomialssn 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.

Generalized cardinal B-splines: Stability, linear independence, and appropriate scaling matrices

Abstract: Generalized cardinal B-splines are defined as convolution products of characteristic functions of self-affine lattice tiles with respect to a given integer scaling matrix. By construction, these generalized splines are refinable functions with respect to the scaling matrix and therefore they can be used to define a multiresolution analysis and to construct a wavelet basis. In this paper, we study the stability and linear independence properties of the integer translates of these generalized spline functions. Moreover, we give a characterization of the scaling matrices to which the construction of the generalized spline functions can be applied.

Lp-Error Bounds for Hermite Interpolation and the Associated Wirtinger Inequalities

Abstract: The B-spline representation for divided differences is used, for the first time, to provide Lp-bounds for the error in Hermite interpolation, and its derivatives, thereby simplifying and improving the results to be found in the extensive literature on the problem. These bounds are equivalent to certain Wirtinger inequalities.

Expansions in Jacobi Polynomials of Negative Order

Abstract: We extend an equiconvergence theorem for Jacobi expansions which is well known for α, β > -1 to the general case of all α, \(\beta \in {\Bbb R}\) .

Partition of unity and density: A counterexample

Abstract: By providing a counterexample we show that there exists a shift-invariant space S generated by a piecewise linear function such that the union of the corresponding scaled spaces S h (h > 0) is dense in C0.R 2 / but S does not contain a stable and locally supported partition of unity. This settles a question raised by C. de Boor and R. DeVore a decade ago.

Leading Coefficients and Extreme Points of Homogeneous Polynomials

Abstract: Recently, H. Hakopian proved that the squares of a bivariate homogeneous polynomial and of its gradient have, in general, the same set of maximum points on the sphere. A generalization of this result mentioned can be gained by a method O. D. Kellogg used years ago in the estimate of some coefficient functionals.

Uniqueness of Best Parametric Interpolation by Cubic Spline Curves

Abstract: Best parametric spline interpolation extends and refines the classical spline problem of best interpolation to \( {(*)}\quad\quad \inf_{\underline t} \inf_{\underline f}\left\{\int^1_0 \Vert \underline f^{(k)} (t) \Vert^2 \,dt \colon \ \underline f(t_i) = \underline y_i, 1 \leq i \leq n \right\} \) Here \( \underline t \colon \ 0 = t_1 < \cdots < t_n = 1 \) denotes a sequence of nodes and \(\underline y_i \) data in \( \mbox{\footnotesize\bf R}^d \) with \( \underline y_i ot = \underline y_{i+1} \) The \( \mbox{\footnotesize\bf R}^d \) -valued functions \( \underline f(t) \) lie componentwise in the Sobolev space \( L^k_2 (0,1) \) and || || denotes the Euclidean norm in R^d This problem has been posed by H J Toepfer (1981) and independently by S Marin (1984) who showed existence and uniqueness of the solution for d = 1 and k = 2 Here we extend this result to the general parametric case \( d \geq 1\) To this end, the problem is rewritten as a (nonclassical) saddle-point problem It is concave for a part of the variables but not convex for the other one Nevertheless, we can prove the existence of the unique solution of it and hence of (*) By an alternative approach via degree theory we prove further that there is at most one local strict minimum of (*) Finally, we discuss the relation to other problems of best interpolation where the minimization is performed with respect to the curvature or a certain class of β-splines