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

We show that the limiting polygon generated by the dimension elevation algorithm with respect to the Müntz space span(1,tr1,tr2,trm,. . .), with 0 < _r1 < _r2 < ⋯ < _{r m} < ⋯ and _{lim n →∞r n} = ∞, over an interval [a, b] ⊂ ] 0, ∞ [ converges to the underlying Chebyshev-Bézier curve if and only if the Müntz condition ∑i=1∞1ri=∞ is satisfied. The surprising emergence of the Müntz 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.