Chebyshev Blossom in Muntz Spaces: Toward Shaping with Young Diagrams

TL;DR

This paper introduces a combinatorial approach to Chebyshev blossom in Muntz spaces using Schur functions, leading to elegant algorithms for shape design and basis computation.

Contribution

It provides explicit formulas and algorithms for Chebyshev blossom and basis construction in Muntz spaces, utilizing Young diagrams and Schur functions.

Findings

Explicit Chebyshev blossom expressions in Muntz spaces.

Efficient algorithms for basis and dimension elevation.

Application to free-form shape design schemes.

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 Muntz 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 Muntz spaces in term of Schur functions. We derive an explicit expression of the Chebyshev-Bernstein basis via an inductive argument on nested Muntz spaces. We also reveal a simple algorithm for the dimension elevation process. Free-form design schemes in Muntz spaces with Young diagrams as shape parameter will be discussed.