Reproducing Polynomials with B-Splines

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A B-Spline of order is known to be able to reproduce any polynomial up to order [1]:

In words, a proper linear combination of shifted versions of a B-Spline can reproduce any polynomial up to order . This is needed for different applications, for example, for the Sampling at Finite Rate of Innovation (FRI) framework. In this case any kernel reproducing polynomials (that is, satisfying the Strang-Fix conditions) can be used. However, among all possible kernels, the B-Splines have the smallest possible support.

An important question is how to obtain the coefficients for the reproduction-formula. In this small article, I describe one way.

  1. I.J. Schoenberg: Cardinal interpolation and spline functions