A universal formula for generalized cardinal B-splines

Abstract

We introduce a universal and systematic way of defining a generalized B-spline based on a linear shift-invariant (LSI) operator L (a.k.a. Fourier multiplier). The generic form of the B-spline is β L = L d L − 1 δ where L − 1 δ is the Green's function of L and where L d is the discretized version of the operator that has the smallest-possible null space. The cornerstone of our approach is a main construction of L d in the form of an infinite product that is motivated by Weierstrass' factorization of entire functions. We show that the resulting Fourier-domain expression is compatible with the construction of all known B-splines. In the special case where L is the derivative operator (linked with piecewise-constant splines), our formula is equivalent to Euler's celebrated decomposition of sinc ( x ) = sin ⁡ ( π x ) π x into an infinite product of polynomials. Our main challenge is to prove convergence and to establish continuity results for the proposed infinite-product representation. The ultimate outcome is the demonstration that the generalized B-spline β L generates a Riesz basis of the space of cardinal L-splines, where L is an essentially arbitrary pseudo-differential operator.