Reproducing Polynomials with B-Splines

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


\sum_{n \in \mathbb{Z}} c_{m,n} \beta_N (t - n) = t^m

In words, a proper linear combination of shifted versions of a B-Spline can reproduce any polynomial up to order N. This is needed for different applications, for example, for the Sampling at Finite Rate of Innovation (FRI) framework. In this case any kernel \varphi 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 c_{m,n} for the reproduction-formula. In this small article, I describe one way.

  1. I.J. Schoenberg: "Cardinal interpolation and spline functions", J. Approx. Theory volume 2, pp. 167-206, 1969
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