Abstract
We show that the generalized Bernstein bases in Müntz spaces defined by Hirschman and Widder (1949) and extended by Gelfond (1950) can be obtained as pointwise limits of the Chebyshev-Bernstein bases in Müntz spaces with respect to an interval [a,1] as the positive real number a converges to zero. Such a realization allows for concepts of curve design such as de Casteljau algorithm, blossom, dimension elevation to be transferred from the general theory of Chebyshev blossoms in Müntz spaces to these generalized Bernstein bases that we termed here as Gelfond-Bernstein bases. The advantage of working with Gelfond-Bernstein bases lies in the simplicity of the obtained concepts and algorithms as compared to their Chebyshev-Bernstein bases counterparts.
| Original language | English |
|---|---|
| Pages (from-to) | 199-225 |
| Number of pages | 27 |
| Journal | Computer Aided Geometric Design |
| Volume | 30 |
| Issue number | 2 |
| DOIs | |
| State | Published - 2013 |
| Externally published | Yes |
Bibliographical note
Funding Information:This work was partially supported by the MEXT Global COE project at Osaka University, Japan. The authors would like to thank the anonymous referees for their careful revision of the manuscript and helpful comments and also to thank the anonymous referee for pointing out the optimality of the Gelfond–Bernstein bases.
Keywords
- Chebyshev blossom
- Chebyshev-Bernstein basis
- Gelfond-Bézier curve
- Geometric design
- Müntz spaces
- Schur functions
- Young diagrams
ASJC Scopus subject areas
- Modeling and Simulation
- Automotive Engineering
- Aerospace Engineering
- Computer Graphics and Computer-Aided Design