Lagrange Interpolatory Subdivision Schemes in Chebyshev Spaces

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.