Given scattered data in IRS, interpolation from a dilated box spline space SM(2~k) is always possible for a fine enough scaling. For example, for the Lagrange function of a point 6 one could take any shifted dilate M(2k -j) which is nonzero at 6 and zero at the other interpolation points However, the resulting interpolant, though smooth (and local), will consist of a set of "bumps", and so by any reasonable measure provides a poor representation of the shape of the underlying function. On the other hand, it is possible to choose a space of interpolants which contains some M(2fc -j) of arbitrarily large support. But the resulting methods are increasingly less local, and in general still require some splines with a much higher level of dilation. Here we provide a multilevel method which constructs a space of interpolants by taking as many splines as possible from a given dilation level, then as many from the next (higher) dilation level, and so forth The choice at each level is made using the suggestion of [18], which is based on the Riesz representation theorem. This requires an inner product on the ground space SM> and the higher levels SM(2fe ) e SM(2 k-1-), k = 1,2, The inner products used here involve the box spline coefRcients, and prewavelet coefficients of [15], respectively, and are norm equivalent to ||·||L2(Rs)-^sThese lead to a scheme which is easily implemented, and numerically stable. Previously, box spline interpolants have been considered only for data on a regular grid.
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