A Muntz Type Theorem for a Family of Corner Cutting Schemes

TL;DR

This paper establishes a Muntz type theorem for a family of corner cutting schemes viewed as Gelfond-Bezier curve dimension elevation, linking convergence conditions to a classical density criterion.

Contribution

It introduces a novel connection between Muntz's theorem and the convergence of corner cutting schemes via Gelfond-Bezier curves.

Findings

Muntz condition characterizes convergence of control polygons

Connection between density of Chebyshev spaces and algorithm convergence

New perspective on corner cutting schemes through classical analysis

Abstract

By identifying a family of corner cutting schemes as a dimension elevation process of Gelfond-Bezier curves, we give a Muntz type condition for the convergence of the generated control polygons to the underlying curve. 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.