Perfect spline approximation

Our study of perfect spline approximation reveals: (i) it is closely related to ∑Δ modulation used in one-bit quantization of bandlimited signals. In fact, they share the same recursive formulae, although in different contexts; (ii) the best rate of approximation by perfect splines of order r with equidistant knots of mesh size h is h^r-1. This rate is optimal in the sense that a function can be approximated with a better rate if and only if it is a polynomial of degree <r. The uniqueness of best approximation is studied, too. Along the way, we also give a result on an extremal problem, that is, among all perfect splines with integer knots on ℝ, (multiples of) Euler splines have the smallest possible norms.