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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Approximation by smooth multivariate splines
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by C. de Boor and R. DeVore PDF
Trans. Amer. Math. Soc. 276 (1983), 775-788 Request permission


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.
    C. de Boor and G. Fix, Spline approximation by quasi-interpolants, J. Approx. Theory 7 (1973), 19-45.
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Additional Information
  • © Copyright 1983 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 276 (1983), 775-788
  • MSC: Primary 41A15; Secondary 41A25, 41A63
  • DOI:
  • MathSciNet review: 688977