Asymptotically Optimal Approximation and Numerical Solutions of Diierential Equations Asymptotically Optimal Approximation and Numerical Solutions of Diierential Equations

Given nite subset J IR n , and a point 2 IR n , we study in this paper the possible convergence, as h ! 0, of the coeecients in least-squares approximation to f(+hh) from the space spanned by (f(+ hj) j2J. We invoke thèleast solution of the polynomial interpolation problem' to show that the coeecient do converge for a generic J and , provided that the underlying function f is suuciently smooth. Moreover, in certain cases (e.g., in the case J for a rectangular mesh), the limit of the least-squares coeecients are shown to be independent of f, and are characterized by their polynomial accuracy. Finally, we employ a diierent argument to show that convergence of the least squares coeecients occurs also for a certain class of functions which are not \suuciently smooth". The above study is relevant to the problem of selecting an optimal diierencing scheme for solving PDE's, a connection that is brieey discussed as well.