Polynomial approximation, local polynomial convexity, and degenerate CR singularities

We begin with the following question: given a closed disc and a complex-valued function , is the uniform algebra on generated by z and F equal to ? When F∈C¹(D), this question is complicated by the presence of points in the surface that have complex tangents. Such points are called CR singularities. Let p∈S be a CR singularity at which the order of contact of the tangent plane with S is greater than 2; i.e. a degenerate CR singularity. We provide sufficient conditions for S to be locally polynomially convex at the degenerate singularity p. This is useful because it is essential to know whether S is locally polynomially convex at a CR singularity in order to answer the initial question. To this end, we also present a general theorem on the uniform algebra generated by z and F, which we use in our investigations. This result may be of independent interest because it is applicable even to non-smooth, complex-valued F.     

On stability of some linear and nonlinear delay differential equations

New explicit conditions of exponential stability are obtained for the nonautonomous equation with several delays

     

Ambiguous chance constrained problems and robust optimization

In this paper we study ambiguous chance constrained problems where the distributions of the random parameters in the problem are themselves uncertain. We focus primarily on the special case where the uncertainty set of the distributions is of the form where ρ_p denotes the Prohorov metric. The ambiguous chance constrained problem is approximated by a robust sampled problem where each constraint is a robust constraint centered at a sample drawn according to the central measure The main contribution of this paper is to show that the robust sampled problem is a good approximation for the ambiguous chance constrained problem with a high probability. This result is established using the Strassen-Dudley Representation Theorem that states that when the distributions of two random variables are close in the Prohorov metric one can construct a coupling of the random variables such that the samples are close with a high probability. We also show that the robust sampled problem can be solved efficiently both in theory and in practice. Research partially supported by NSF grant CCR-00-09972. Research partially supported by NSF grants CCR-00-09972, DMS-01-04282, and ONR grant N000140310514.     

The Averaged Robbins – Monro Method for Linear Problems in a Banach Space

We consider a recursive method of Robbins–Monro type to solve the linear problem Ax=V in a Banach space. The bounded linear operator A and the vector V are assumed to be observable with some noise only. According to Polyak and Ruppert we use gains converging to zero slower than 1/n and take the average of the iterates as an estimator for the solution of the linear problem. Under weak conditions on the noise processes almost sure and distributional invariance principles are shown.     

Finite‐difference approximation for the u(k)‐derivative with O(hM−k+1) accuracy: An analytical expression

An approximation of function u(x) as a Taylor series expansion about a point x₀ at M points x_i, ～ i = 1,2,…,M is used where x_i are arbitrary‐spaced. This approximation is a linear system for the derivatives u^(k) with an arbitrary accuracy. An analytical expression for the inverse matrix A ^?1 where A = [A_ik] = (x_i ? x₀)^k is found. A finite‐difference approximation of derivatives u^(k) of a given function u(x) at point x₀ is derived in terms of the values u(x_i). © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

The solvability conditions for the inverse eigenproblems of symmetric and generalized centro-symmetric matrices and their approximations

Let ∥ · ∥ be the Frobenius norm of matrices. We consider (I) the set S_E of symmetric and generalized centro-symmetric real n × n matrices R_n with some given eigenpairs (λ_j, q_j) (j = 1, 2, … , m) and (II) the element in S_E which minimizes for a given real matrix R^∗. Necessary and sufficient conditions for S_E to be nonempty are presented. A general form of elements in S_E is given and an explicit expression of the minimizer is derived. Finally, a numerical example is reported.     

"Push-the-Error" Algorithm for Nonlinear n-Term Approximation

This paper is concerned with further developing and refining the analysis of a recent algorithmic paradigm for nonlinear approximation, termed the "Push-the-Error" scheme. It is especially designed to deal with L_∞-approximation in a multilevel framework. The original version is extended considerably to cover all commonly used multiresolution frameworks. The main conceptually new result is the proof of the quasi-semi-additivity of the functional N(ε) counting the number of terms needed to achieve accuracy ε. This allows one to show that the improved scheme captures all rates of best n-term approximation.     

Lower Bounds for n-Term Approximations of Plane Convex Sets and Related Topics

In this paper, we establish lower bounds for n-term approximations in the metric of L ²⁽I ² ) of characteristic functions of plane convex subsets of the square I ² with respect to arbitrary orthogonal systems. It is shown that, as n, these bounds cannot decrease more rapidly than .     

Approximation Properties of the Operators ⋎n+2r(f) and of Their Discrete Analogs

This paper is devoted to the study of the approximation properties of linear operators which are partial Fourier--Legendre sums of order n with 2r terms of the form _k=1 ^2r a_kP_n+k(x) added; here P _m(x) denotes the Legendre polynomial. Due to this addition, the linear operators interpolate functions and their derivatives at the endpoints of the closed interval [-1,1], which, in fact, for r= = 1 allows us to significantly improve the approximation properties of partial Fourier--Legendre sums. It is proved that these operators realize order-best uniform algebraic approximation of the classes of functions and A _q (B). With the aim of the computational realization of these operators, we construct their discrete analogs by means of Chebyshev polynomials, orthogonal on a uniform grid, also possessing nice approximation properties.