Moment Preserving Local Spline Projection Operators

Abstract

This article describes an elementary construction of a dual basis for nonuniform B-splines that is local, $$L^\infty $$-stable, and reproduces polynomials of any prescribed degree. This allows one to define local projection operators with near-optimal approximation properties in any $$L^q$$, $$1 \le q \le \infty $$, and high order moment preserving properties. As the dual basis functions share the same piecewise polynomial structure as the underlying splines, simple quadrature formulas can be used to compute the projected spline coefficients.