Asymptotically Optimal Approximation and Numerical Solutions of Differential Equations

Given finite subset J ⊂ IR, and a point λ ∈ IR, we study in this paper the possible convergence, as h→ 0, of the coefficients in least-squares approximation to f(·+hλ) from the space spanned by (f(·+ hj)j∈J . We invoke the ‘least solution of the polynomial interpolation problem’ to show that the coefficient do converge for a generic J and λ, provided that the underlying function f is sufficiently smooth. Moreover, in certain cases (e.g., in the case J ∪λ for a rectangular mesh), the limit of the least-squares coefficients are shown to be independent of f , and are characterized by their polynomial accuracy. Finally, we employ a different argument to show that convergence of the least squares coefficients occurs also for a certain class of functions which are not “sufficiently smooth”. The above study is relevant to the problem of selecting an optimal differencing scheme for solving PDE’s, a connection that is briefly discussed as well. This work was supported by the National Science Foundation under Grants DMS-9224748 and DMS-9626319, and by the U.S. Army Research Office under Contract DAAH04-95-1-0089.