Dimension elevation in Muntz spaces: A new emergence of the Muntz condition

TL;DR

This paper establishes that the convergence of the dimension elevation algorithm in Muntz spaces to Chebyshev-Bézier curves depends precisely on the Muntz condition, linking space density to geometric approximation.

Contribution

It proves that the Muntz condition is necessary and sufficient for the convergence of the dimension elevation algorithm in Muntz spaces to Chebyshev-Bézier curves.

Findings

Convergence occurs if and only if the Muntz condition holds.

The Muntz condition relates to the density of the space in approximation theory.

Open problem: convergence without monotonicity or positivity constraints.

Abstract

We show that the limiting polygon generated by the dimension elevation algorithm with respect to the \muntz space $span(1,t^{r_1},t^{r_2},...,t^{r_m},...)$, with $0 < r_1 < r_2 < ... < r_m < ...$ and $\lim_{n\to\infty}r_n = \infty$, over an interval $[a,b]\subset]0,\infty[$ converges to the underlying Chebyshev-B\'ezier curve if and only if the \muntz condition $\sum_{i=1}^{\infty} \frac{1}{r_i} = \infty$ is satisfied. The surprising emergence of the \muntz condition in the problem raises the question of a possible connection between the density questions of nested Chebyshev spaces and the convergence of the corresponding dimension elevation algorithms. The question of convergence with no condition of monotonicity or positivity on the pairwise distinct real numbers $r_i$ remains an open problem.