Generating Functions for Uniform B-Splines

Abstract

We derive a closed formula for the generating functions of the uniform B-splines. We begin by constructing a PDE for these generating functions starting from the de Boor recurrence. By solving this PDE, we find that we can express these generating functions explicitly as sums of polynomials times exponentials. Using these generating functions, we derive some known identities, including the Schoenberg identity, the two term formula for the derivatives in terms of B-splines of lower degree, and the partition of unity property. We also derive several new identities for uniform B-splines not previously available from classical methods, including formulas for sums and alternating sums, for moments and reciprocal moments, and for Laplace transforms and convolutions with monomials.