A CHARACTERIZATION OF MULTIVARIATE QUASI-INTERPOLATION FORMULAS AND ITS APPLICATIONS

Let φ{symbol} be a compactly supported function on ℝs and S (φ{symbol}) the linear space with generator φ{symbol}; that is, S (φ{symbol}) is the linear span of the multiinteger translates of φ{symbol}. It is well known that corresponding to a generator φ{symbol} there are infinitely many quasi-interpolation formulas. A characterization of these formulas is presented which allows for their direct calculation in a variety of forms suitable to particular applications, and in addition, provides a clear formulation of the difficult problem of minimally supported quasi-interpolants. We introduce a generalization of interpolation called μ-interpolation and a notion of higher order quasi-interpolation called μ-approximation. A characterization of μ-approximants similar to that of quasi-interpolants is obtained with similar applications. Among these applications are estimating least-squares approximants without matrix inversion, surface fitting to incomplete or semi-scattered discrete data, and constructing generators with one-point quasi-interpolation formulas. It will be seen that the exact values of the generator φ{symbol} at the multi-integers ℤs facilitates the above study. An algorithm to yield this information for box splines is discussed. © 1990 Springer-Verlag.