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On two polynomial spaces associated with a box spline. (English) Zbl 0678.41009
Summary: The polynomial space \({\mathcal H}\) in the span of the integer translates of a box spline M admits a well-known characterization as the joint kernel of a set of homogeneous differential operators with constant coefficients. The dual space \({\mathcal H}^*\) has a convenient representation by a polynomial space \({\mathcal P}\), explicitly known, which plays an important role in box spline theory as well as in multivariate polynomial interpolation. We characterize the dual space \({\mathcal P}\) as the joint kernel of simple differential operators, each one a power of a directional derivative. Various applications of this result to multivariate polynomial interpolation, multivariate splines and duality between polynomial and exponential spaces are discussed.

41A15 Spline approximation
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