Approximation by Smooth Multivariate Splines

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.