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

作者：Ait Haddou Rachid^*

来源：Journal of Approximation Theory, 2014, 181: 6-17.

DOI：10.1016/j.jat.2014.01.006

• Muntz spaces
• Chebyshev blossoming
• Dimension elevation
• Chebyshev-Bernstein bases
• Gelfond-Bernstein bases
• Schur functions
• Chebyshev-Bezier curves

We show that the limiting polygon generated by the dimension elevation algorithm with respect to the Muntz space span(1, t(r1), t(r2), t(rm) , . . .), with 0 %26lt; r(1) %26lt; r(2), %26lt; . . . %26lt; r(m) %26lt; . . . and lim(n -%26gt;infinity)r(n) = infinity, over an interval [a, b] subset of]0, infinity[ converges to the underlying Chebyshev-Bezier curve if and only if the Muntz condition Sigma(infinity)(i=1) 1/r(i) = infinity 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.

• 出版日期2014-5