Chebyshev blossoming in Müntz spaces: toward shaping with young diagrams

R. Ait-Haddou, Y. Sakane, T. Nomura
Journal of Computational and Applied Mathematics, volume 247, pp. 172-208, (2013)

Chebyshev blossoming in Müntz spaces: toward shaping with young diagrams

Keywords

Extended Chebyshev systems, Chebyshev blossom, Computer aided design, Chebyshev-Bernstein basis, Schur functions, Young diagrams

Abstract

​The notion of a 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 a 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 terms of Schur functions. We derive an explicit expression for the Chebyshev-Bernstein basis via an inductive argument on nested Muntz spaces. We also reveal a simple algorithm for dimension elevation. Free-form design schemes in Muntz spaces with Young diagrams as shape parameters are discussed.

Code

DOI: 10.1016/j.cam.2013.01.009

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