Chebyshev blossoming in Müntz spaces: Toward shaping with Young diagrams

Authors:
Rachid Ait-Haddou;Yusuke Sakane;Taishin Nomura
Affiliations:
King Abdullah University of Science and Technology, Thuwal, Saudi Arabia and The Center of Advanced Medical Engineering and Informatics, Osaka University, 560-8531 Osaka, Japan;Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, 560-0043 Osaka, Japan;The Center of Advanced Medical Engineering and Informatics, Osaka University, 560-8531 Osaka, Japan and Department of Mechanical Science and Bioengineering, Graduate School of Engineering Science, ...
Venue:
Journal of Computational and Applied Mathematics
Year:
2013

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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.