Minimum Support Interpolators with Optimum Approximation Properties

We investigate the functions of given approximation order L that have the smallest support. Those are shown to be linear combinations of the B-spline of degree L-1 and its L-1 first derivatives. We then show how to find the functions that minimize the asymptotic approximation constant among this finite dimension space; in particular, a tractable induction relation is worked out. Using these functions instead of splines, we observe that the approximation error is dramatically reduced, not only in the limit when the sampling step tends to zero, but also for higher values up to the Shannon rate. Finally, we show that those optimal functions satisfy a scaling equation, although less simple than the usual two-scale difference equation.