Quasi-interpolation projectors for box splines

Scattered data interpolation by box splines

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...

Constrained Interpolation via Cubic Hermite Splines

Introduction In industrial designing and manufacturing, it is often required to generate a smooth function approximating a given set of data which preserves certain shape properties of the data such as positivity, monotonicity, or convexity, that is, a smooth shape preserving approximation.  It is assumed here that the data is sufficiently accurate to warrant interpolation, rather than least ...

Interpolation, box splines, and lattice points in zonotopes

Given a finite list of vectors X ⊆ R, one can define the box spline BX . Box splines are piecewise polynomial functions that are used in approximation theory. They are also interesting from a combinatorial point of view and many of their properties solely depend on the structure of the matroid defined by the list X . The support of the box spline is the zonotope Z(X). We show that if the list X...

A Natural Formulation of Quasi-interpolation by Multivariate Splines

A quasi-interpolation formula based on discrete function values is given in the form of a multivariate spline series that yields the local approximation order characterized by the Fix-Strang conditions. This formula can be considered as a partial sum of the formal Neumann series expansion of the formal interpolation operator, and hence, justifies that "quasi-interpolation" is indeed an appropri...

Using multiquadric quasi-interpolation for solving Kawahara equation