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

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-Bézier 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.