Abstract
We develop an efficient approach to the analysis of Lagrange interpolatory subdivision schemes based on Extended Chebyshev spaces of any even dimension. In general, such schemes are non-uniform and non-stationary. The study confirms and extends some ideas concerning more generally the analysis of non-regular subdivision schemes already presented in earlier papers. One crucial step consists in finding (non-regular) grids naturally adapted to the initial scheme in view of defining its derived schemes, a change of grid being possibly necessary for each additional order of smoothness considered. Surprisingly, it may be the case that the natural grids are non-nested, even though the initial scheme is interpolatory. This is so in particular for Chebyshevian Lagrange interpolatory schemes, for which the natural grids are defined in terms of Chebyshevian divided differences. Comparison of the corresponding successive derived schemes with their polynomial counterparts enables us to show that they have similar behaviours.
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Acknowledgments
I mainly developed the ideas presented in this article during a short stay in Tel Aviv in January 2010. I am thus extremely grateful to Nira Dyn and David Levin for the invitation and the stimulating discussions we had together, without which this work would not have been possible.
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Communicated by Albert Cohen.
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Mazure, ML. Lagrange Interpolatory Subdivision Schemes in Chebyshev Spaces. Found Comput Math 15, 1035–1068 (2015). https://doi.org/10.1007/s10208-014-9209-9
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DOI: https://doi.org/10.1007/s10208-014-9209-9
Keywords
- Subdivision
- Lagrange interpolatory schemes
- Non-nested grids
- Extended Chebyshev spaces
- Chebyshevian divided differences
- Blossoms