Numerische Mathematik

, Volume 57, Issue 1, pp 105–121 | Cite as

A characterization of multivariate quasi-interpolation formulas and its applications

  • Charles K. Chui
  • Harvey Diamond


Let ϕ be a compactly supported function on ℝ s 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 ℤ s facilitates the above study. An algorithm to yield this information for box splines is discussed.

Subject Classifications

AMS(MOS): 41A15, 41A25, 41A63 


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Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Charles K. Chui
    • 1
  • Harvey Diamond
    • 2
  1. 1.Department of MathematicsTexas A & M UniversityCollege StationUSA
  2. 2.Department of MathematicsWest Virginia UniversityMorgantownUSA

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