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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Approximation by smooth multivariate splines

Authors: C. de Boor and R. DeVore
Journal: Trans. Amer. Math. Soc. 276 (1983), 775-788
MSC: Primary 41A15; Secondary 41A25, 41A63
MathSciNet review: 688977
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Abstract: The degree of approximation achievable by piecewise polynomial functions of given total order on certain regular grids in the plane is shown to be adversely affected by smoothness requirements--in stark contrast to the univariate situation. For a rectangular grid, and for the triangular grid derived from it by adding all northeast diagonals, the maximum degree of approximation (as the grid size $ 1/n$ goes to zero) to a suitably smooth function is shown to be $ O({n^{- \rho - 2}})$ in case we insist that the approximating functions are in $ {C^\rho}$. This only holds as long as $ \rho \leqslant (r - 3)/2$ and $ \rho \leqslant (2r - 4)/3$, respectively, with $ r$ the total order of the polynomial pieces. In the contrary case, some smooth functions are not approximable at all. In the discussion of the second mesh, a new and promising kind of multivariate $ {\text{B}}$-spline is introduced.

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Keywords: Multivariate, splines, piecewise polynomial, smoothness, degree of approximation, $ {\text{B}}$-splines
Article copyright: © Copyright 1983 American Mathematical Society