Simultaneous zero-free approximation and universal optimal polynomial approximants

Degree of Simultaneous Coconvex Polynomial Approximation

Let f 2 C 1 ?1; 1] change its convexity nitely many times in the interval, say s times, at Y s : ?1 < y s < < y 1 < 1. We estimate the degree of simultaneous approximation of f and its derivative by polynomials of degree n, which change convexity exactly at the points Y s , and their derivatives. We show that provided n is suuciently large, depending on the location of the points Y s , the rate...

Polynomial Multivariate Density Estimates and Approximants

It is often the case that the exact moments of a continuous distribution whose support is confined to a closed interval can be explicitly determined, while its exact density function either does not lend itself to numerical evaluation or proves to be mathematically intractable. A representation of a least-squares approximating polynomial which does not rely on orthogonal polynomials, is obtaine...

Simultaneous Approximation by a Polynomial and Its Derivatives

Theorem. If fo is a continuous function on a compact set C without interior and with connected complement of the complex plane, and if fi, • • ■ i fn are continuous functions on a compact totally disconnected subset E of C, then there exists a sequence {pi} of polynomials such that pi-^fo uniformly on C as i—* , and pim—*fk uniformly on E as i—>», for lslk^n, where pi[k] is the kth derivativ...

Simultaneous Approximation of Zero by Values of Integral Polynomials

Polynomial Bases for Optimal Worst Case Approximation of Nonlinearities

Several identification and control problems present nonlinearities that cannot be neglected and are often approximated by polynomials. In some previous works optimal set of interpolation nodes that minimizes the uncertainties of the approximation have been derived for the Vandermonde base that, however, can lead to ill-conditioned numerical problems. In this paper the conditions under which pol...