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

Abstract

We show that the limiting polygon generated by the dimension elevation algorithm with respect to the Müntz space span(1, t r1 , t r2 , ..., t rm , ...), with 0 < r 1 < r 2 < ... < 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 1 ri = ∞ 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.

Control point based exact description of curves and surfaces, in extended Chebyshev spaces

“…However, in general, the initial ordinary basis F α,β n can iteratively be appended by new linearly independent functions in infinitely many ways and not every choice of functions leads to a sequence of order elevated control polygons that fulfills the desired convergence property, e.g. in EC Müntz spaces a recent characterization of the required convergence of the dimension/order elevation process can be found in [1]. In order to illustrate the last step of Algorithm 2.1, Fig.…”

“…Let n ≥ 1 be a fixed integer and consider the extended Chebyshev (EC) system F α,β n = ϕ n,i (u) : u ∈ [α, β] n i=0 , ϕ n,0 ≡ 1, − ∞ < α < β < ∞ (1) of basis functions in C n ([α, β]), i.e., by definition [15], for any integer 0 ≤ r ≤ n, any strictly increasing sequence of knot values α ≤ u 0 < u 1 < . .…”

“…Based on homogeneous coordinates and central projection, we also propose an algorithm for the control point (and weight) based exact description of the rational counterpart of these ordinary integral curves. Results will also be extended to the exact representation of families of (hybrid) integral and rational surfaces that are exclusively given in each of their variables by using ordinary EC basis functions of the type (1).…”

Interpolation of G1 Hermite data by C1 cubic-like sparse Pythagorean hodograph splines

“…, r n are consecutive integers, for design algorithms are hardly more complicated that in the polynomial case [29,30,3]. Such spaces are referred to as complete Müntz spaces in [3], see also [6,5]. From now on, we limit ourselves to the cubic-like case n = 3.…”

Approximation by Müntz spaces on positive intervals

“…With these notations, the key-point to prove Theorem 3.1 is the following lemma, for the proof of which we refer to [2], see also [1]. Lemma 3.2 Assume that the Müntz density condition (8) holds.…”