Improvement of a Theorem of Lorentz (1963) and its Generalization to the Multivariate Case

TL;DR

This paper improves Lorentz's theorem by providing a more precise estimate of polynomial approximation with positive coefficients, extending it to multivariate cases and enhancing the understanding of density approximation.

Contribution

It presents an enhanced version of Lorentz's theorem and generalizes it to multivariate cases, offering a better mathematical foundation for density approximation.

Findings

Non-uniform approximation estimates at vertices of [0,1]^d

Enhanced theorem provides more precise approximation characterization

Multivariate generalization supports density approximation methods

Abstract

In this short note we have proved an enhanced version of a theorem of Lorentz [1] and its generalization to the multivariate case which gives a non- uniform estimate of degree of approximation by a polynomial with positive coefficients. The performance of the approximation at the vertices of [0; 1]d is more precisely characterized by the improved result and its multivariate generalization. The latter provides mathematical foundation on which multivariate density approximation by a polynomial with positive coefficients can be established.