  Generalized sampling consists in the recovery of a function $f$, from the
samples of the responses of a collection of linear shift-invariant systems to
the input $f$. The reconstructed function is typically a member of a finitely
generated integer-shift-invariant space that can reproduce polynomials up to a
given degree $M$. While this property allows for an approximation power of
order $(M+1)$, it comes with a tradeoff on the length of the support of the
basis functions. Specifically, we prove that the sum of the length of the
support of the generators is at least $(M+1)$. Following this result, we
introduce the notion of shortest basis of degree $M$, which is motivated by our
desire to minimize the computational costs. We then demonstrate that any basis
of shortest support generates a Riesz basis. Finally, we introduce a recursive
algorithm to construct the shortest-support basis for any multi-spline space.
It provides a generalization of both polynomial and Hermite B-splines. This
framework paves the way for novel applications such as fast derivative sampling
with arbitrarily high approximation power.
