Optimal distribution of knots for tensor-product spline approximation
This paper deals with optimal asymptotic distribution of knots for various types of spline approximations. Known results on the optimal distribution of knots, which asymptotically minimizes the Lp[0, 1] error in spline approximation of a sufficiently smooth function, are extended to the following cases: (1) An optimal distribution for a simultaneous Lp[0, 1] approximation of a finite set of functions, or a set which depends on a continuous parameter, e.g., a bivariate function. (2) An optimal distribution on each axis for L-,[0, 1] tensor-product spline approximation. This results extends to L2[0, \]n , n > 2. A method of knot placement for tensor-product spline approximation is proposed, based on the theoretical optimal distribution.