Computing quasi-interpolants from the B-form of B-splines

In general, for a sufficiently regular function, an expression for the quasi-interpolation error associated with discrete, differential and integral quasi-interpolants can be derived involving a term measuring how well the non-reproduced monomials are approximated. That term depends on some expressions of the coefficients defining the quasi-interpolant, and its minimization has been proposed. However, the resulting problem is rather complex and often requires some computational effort. Thus, for quasi-interpolants defined from a piecewise polynomial function, φ, we propose a simpler minimization problem, based on the Bernstein–Bézier representation of some related piecewise polynomial functions, leading to a new class of quasi-interpolants.