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On a general new class of quasi Chebyshevian splines. (English) Zbl 1232.41008

Extended Chebyshev spaces are natural generalisations of polynomial spaces and have become classical tools in approximation theory. Recently, the author introduced the larger class of quasi extended Chebyshev (QEC) spaces and proved that it is the largest class of spaces with sufficient differentiability which are suitable for design.
The author first develops the general theory of quasi Chebyshevian splines (i.e. splines with all sections in the same QEC space) based on the properties of blossoms, including the existence and optimality of quasi B-spline bases. The main result of the paper states that a general class of splines with sections in different QEC spaces can be viewed as quasi Chebyshevian splines. This bases on the key fact that it is always possible to connect any pair of QEC spaces of dimension two on adjacent intervals so that it forms a global two-dimensional QEC space. Numerous comparisons with the existing literature are provided, showing that the class of quasi Chebyshevian splines includes a large category of interesting and useful splines.

MSC:

41A15 Spline approximation
65D05 Numerical interpolation
65D17 Computer-aided design (modeling of curves and surfaces)
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