B-splines with homogenized knots

The use of homogenized knots for manipulating univariate polynomials by blossoming algorithms is extended to piecewise polynomials. A generalization of the B-spline to homogenized knots is studied. The new B-spline retains the triangular blossoming algorithms for evaluation, differentiation and knot insertion. Moreover, the B-spline is locally supported and a Marsden’s identity exists. Spaces of natural splines and certain polynomial spline spaces with more general continuity properties than ordinary splines have bases of B-splines over homogenized knots. Applications to nonpolynomial splines such as trigonometric and hyperbolic splines are made.