ON CONVOLUTIONS OF B-SPLINES

A smooth approximation to a function f is achieved by convolving f with a smooth function phi. When phi is nonnegative, of unit mean value, compactly supported and has certain symmetry properties, convolving with phi respects the shape properties of the data f such as local positivity, monotonicity and convexity. We study the convolution of f and phi when phi is a univariate B-spline, tenser product B-spline, box spline or simplex spline, and f is a linear combination of the same kind of splines as phi. 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.