Approximation Theoretic Properties Research Articles

Single-exponential bounds for the smallest singular value of Vandermonde matrices in the sub-Rayleigh regime

Following recent interest by the community, the scaling of the minimal singular value of a Vandermonde matrix with nodes forming clusters on the length scale of Rayleigh distance on the complex unit circle is studied. Using approximation theoretic properties of exponential sums, we show that the decay is only single exponential in the size of the largest cluster, and the bound holds for arbitrary small minimal separation distance. We also obtain a generalization of well-known bounds on the smallest eigenvalue of the generalized prolate matrix in the multi-cluster geometry. Finally, the results are extended to the entire spectrum.

Chebyshev polynomials, moment matching, and optimal estimation of the unseen

We consider the problem of estimating the support size of a discrete distribution whose minimum nonzero mass is at least $\frac{1}{k}$. Under the independent sampling model, we show that the sample complexity, that is, the minimal sample size to achieve an additive error of $\varepsilon k$ with probability at least 0.1 is within universal constant factors of $\frac{k}{\log k}\log^{2}\frac{1}{\varepsilon }$, which improves the state-of-the-art result of $\frac{k}{\varepsilon^{2}\log k}$ in [In Advances in Neural Information Processing Systems (2013) 2157–2165]. Similar characterization of the minimax risk is also obtained. Our procedure is a linear estimator based on the Chebyshev polynomial and its approximation-theoretic properties, which can be evaluated in $O(n+\log^{2}k)$ time and attains the sample complexity within constant factors. The superiority of the proposed estimator in terms of accuracy, computational efficiency and scalability is demonstrated in a variety of synthetic and real datasets.

Ellipse-preserving Hermite interpolation and subdivision

We introduce a family of piecewise-exponential functions that have the Hermite interpolation property. Our design is motivated by the search for an effective scheme for the joint interpolation of points and associated tangents on a curve with the ability to perfectly reproduce ellipses. We prove that the proposed Hermite functions form a Riesz basis and that they reproduce prescribed exponential polynomials. We present a method based on Green's functions to unravel their multi-resolution and approximation-theoretic properties. Finally, we derive the corresponding vector and scalar subdivision schemes, which lend themselves to a fast implementation. The proposed vector scheme is interpolatory and level-dependent, but its asymptotic behavior is the same as the classical cubic Hermite spline algorithm. The same convergence properties—i.e., fourth order of approximation—are hence ensured.

TRANSFORMED POLYNOMIALS FOR NONLINEAR AUTOREGRESSIVE MODELS OF THE CONDITIONAL MEAN

This article proposes a flexible set of transformed polynomial functions for modelling the conditional mean of autoregressive processes. These functions enjoy the same approximation theoretic properties of polynomials and, at the same time, ensure that the process is strictly stationary, is ergodic, has fading memory and has bounded unconditional moments. The consistency and asymptotic normality of the least‐squares estimator is easily obtained as a result. A Monte Carlo study provides evidence of good finite sample properties. Applications in empirical time‐series modelling, structural economics and structural engineering problems show the usefulness of transformed polynomials in a wide range of settings.

Multivariate modified Fourier series and application to boundary value problems

In this paper we analyse the approximation-theoretic properties of modified Fourier series in Cartesian product domains with coefficients from both full and hyperbolic cross index sets. We show that the number of expansion coefficients may be reduced significantly whilst retaining comparable error estimates. In doing so we extend the univariate results of Iserles, Norsett and S. Olver. We then demonstrate that these series can be used in the spectral-Galerkin approximation of second order Neumann boundary value problems, which offers some advantages over standard Chebyshev or Legendre polynomial discretizations.

Direct and inverse results in variable Hilbert scales

Variable Hilbert scales are an important tool for the recent analysis of inverse problems in Hilbert spaces, as these constitute a way to describe smoothness of objects other than functions on domains. Previous analysis of such classes of Hilbert spaces focused on interpolation properties, which allows us to vary between such spaces. In the context of discretization of inverse problems, first results on approximation theoretic properties appeared. The present study is the first which aims at presenting such spaces in the context of approximation theory. The authors review and establish direct theorems and also provide inverse theorems, as such are common in approximation theory.

Regularization by projection: Approximation theoretic aspects and distance functions

Abstract The authors study regularization of non-linear inverse problems by projection methods. Non-linearity is controlled by some range invariance assumption. Emphasis is on approximation theoretic properties of the discretization which determine the convergence rates. Instead of using source conditions for the true solution to represent the error, the authors show how distance functions with respect to some benchmark smoothness are able to replace this. Some examples indicate how the results can be applied.

Replacing points by compacta in neural network approximation

It is shown that cartesian product and pointwise-sum with a fixed compact set preserve various approximation-theoretic properties. Results for pointwise-sum are proved for F-spaces and so hold for any normed linear space, while the other results hold in general metric spaces. Applications are given to approximation of L p -functions on the d-dimensional cube, 1⩽ p<∞, by linear combinations of half-space characteristic functions; i.e., by Heaviside perceptron networks.

Approximation and Besov spaces on stratified groups

This paper treats analogues of the classical Bernstein and Jackson theorems in the nonclassical context of the stratified groups. Besov spaces on stratified groups are characterized by their approximation theoretic properties.

Boundary control of parabolic systems: Finite-element approximation

Finite-element approximation of a Dirichlet type boundary control problem for parabolic systems is considered. An approach based on the direct approximation of an input-output semigroup formula is applied. Error estimates inL2[OT; L2(Ω)] andL2[OT; L2(Γ)] norms are derived for optimal state and optimal control, respectively. It turns out that these estimates areoptimal with respect to the approximation theoretic properties.

A sequence of linear polynomial operators and their approximation-theoretic properties

Abstract : The authors construct a rank diminishing algorithm for matrices and adapt it for application to Lagrange interpolation polynomials (L sub n0) (f;x) of degree = or < (n-1). In r steps the authors arrive at the polynomials (L sub nr) (f;x) of degree (n-r-1), which share many approximation theoretic properties with Lagrange interpolation polynomials. (Author)

Triangular elements in the finite element method

For a plane polygonal domain Ω \Omega and a corresponding (general) triangulation we define classes of functions p m ( x , y ) {p_m}(x,y) which are polynomials on each triangle and which are in C ( m ) ( Ω ) {C^{(m)}}(\Omega ) and also belong to the Sobolev space W 2 ( m + 1 ) ( Ω ) W_2^{(m + 1)}(\Omega ) . Approximation theoretic properties are proved concerning these functions. These results are then applied to the approximate solution of arbitrary-order elliptic boundary value problems by the Galerkin method. Estimates for the error are given. The case of second-order problems is discussed in conjunction with special choices of approximating polynomials.