Abstract
Our study of perfect spline approximation reveals: (i) it is closely related to ΣΔ 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.
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