Chebyshev blossoming in Müntz spaces: Toward shaping with Young diagrams

Abstract

The notion of blossom in extended Chebyshev spaces offers adequate generalizations and extra-utilities to the tools for free-form design schemes. Unfortunately, such advantages are often overshadowed by the complexity of the resulting algorithms. In this work, we show that for the case of Müntz spaces with integer exponents, the notion of Chebyshev blossom leads to elegant algorithms whose complexities are embedded in the combinatorics of Schur functions. We express the blossom and the pseudo-affinity property in Müntz spaces in term of Schur functions. We derive an explicit expression of the Chebyshev-Bernstein basis via an inductive argument on nested Müntz spaces. We also reveal a simple algorithm for the dimension elevation process. Free-form design schemes in Müntz spaces with Young diagrams as shape parameter will be discussed.

Polynomial degree reduction in the discrete $$L_2$$ L 2 -norm equals best Euclidean approximation of h-Bézier coefficients

“…Relations (6.8) are not valid for degree raising, or more appropriately, for dimension raising matrices of Gelfond-Bézier curves [2,4] or Müntz -Bézier curves in general [3]. This may explain to some extent the lack of an analogous result to Corollary 3.1 for Chebyshev spaces other than the polynomial ones.…”

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

“…With the above definitions, we can prove the following theorem [2] Theorem 3. The Chebyshev-Bernstein basis (B n 0,Λn , ..., B n n,Λn ) over an interval…”

“…The strategy for the proof of Theorem 1: For nested Müntz spaces, the analytical form of ξ i in (1) can be expressed in terms of a quotient of generalized Schur functions that depend on the interval parameters a and b [2]. Iterating the dimension elevation process using these quotients leads to complicated expressions that hinder a direct proof of Theorem 1.…”

A unifying structure for polar forms and for Bernstein Bézier curves

“…Typically one keeps the symmetry property, but the other two axioms are malleable. Polar forms for Chebyshev spaces have been constructed by replacing the multi-affine property by a pseudo-affine property (see [1,10,12]); polar forms for quantum Bernstein polynomials and quantum B-splines have been developed by replacing the diagonal property by a quantum diagonal property (see [16,17,15]). …”

“…, x l+n }, where l is non-negative integer. One of the main interests in Müntz spaces that contain constants is that constants make it possible to use all the classical geometric design algorithms for polynomials after appropriately elevating the dimension of the Müntz space (see [11,1]). …”