Reproducing Polynomials with B-Splines
A B-Spline of order is known to be able to reproduce any polynomial up to order :
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.
- ↑ I.J. Schoenberg: Cardinal interpolation and spline functions