On convolutions of B-splines

Abstract

A smooth approximation to a function ƒ is achieved by convolving ƒ with a smooth function φ. When φ is nonnegative, of unit mean value, compactly supported and has certain symmetry properties, convolving with φ respects the shape properties of the data ƒ such as local positivity, monotonicity and convexity. We study the convolution of ƒ and φ when φ is a univariate B-spline, tensor product B-spline, box spline or simplex spline, and ƒ is a linear combination of the same kind of splines as φ. In terms of divided differences and blossoms, we express the convolution of univariate splines over nonuniform knots as linear combinations of B-splines. This conversion can be carried out by a stable recurrence.