Summary
Let ϕ be a compactly supported function on ℝs andS (ϕ) the linear space withgenerator ϕ; that is,S (ϕ) is the linear span of the multiinteger translates of ϕ. It is well known that corresponding to a generator ϕ 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 at the multi-integers ℤs facilitates the above study. An algorithm to yield this information for box splines is discussed.
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Supported by the National Science Foundation and the U.S. Army Research Office
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Chui, C.K., Diamond, H. A characterization of multivariate quasi-interpolation formulas and its applications. Numer. Math. 57, 105–121 (1990). https://doi.org/10.1007/BF01386401
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DOI: https://doi.org/10.1007/BF01386401