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. © 2014 Elsevier Inc.
|Original language||English (US)|
|Number of pages||12|
|Journal||Journal of Approximation Theory|
|State||Published - May 2014|
ASJC Scopus subject areas
- Applied Mathematics
- Numerical Analysis