# Reproducing Polynomials with B-Splines

(Unterschied zwischen Versionen)
A B-Spline of order $N$ is known to be able to reproduce any polynomial up to order $N$:
$\sum_{n \in \mathbb{Z}} c_{m,n} \beta_N (t - n) = t^m$
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.