In the study of approximation order, particularly in a multivariable setting, quasi-interpolants have played a major role. This report points out some limitations of quasi-interpolants and describes some recent results on approximation order obtained without the benefit of the quasi-interpolant idea. §1. Approximation Order In most general terms, “approximation order” is defined as follows. Def...

We introduce a general definition of refinable Hermite interpolants and investigate their general properties. We also study a notion of symmetry of these refinable interpolants. Results and ideas from the extensive theory of general refinement equations are applied to obtain results on refinable Hermite interpolants. The theory developed here is constructive and yields an easyto-use constructio...

In this paper we first revisit a classical problem of computing variational splines. We propose to compute local variational splines in the sense that they are interpolatory splines which minimize the energy norm over a subinterval. We shall show that the error between local and global variational spline interpolants decays exponentially over a fixed subinterval as the support of the local vari...

Given scattered data in IR, interpolation from a dilated box spline space SM (2 ·) is always possible for a fine enough scaling. For example, for the Lagrange function of a point θ one could take any shifted dilate M(2 · −j) which is nonzero at θ 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...

In this paper, the expression of Weierstrass operators as differential operators on polynomials is used for the construction of associated quasi-interpolants. Then the convergence properties of these operators are studied. AMS Classification: 41A35.