sin(ωx) Can Approximate Almost Every Finite Set of Samples

Abstract

Consider a set of points (x1, y1), . . . , (xn, yn) with distinct 0 ≤ xi ≤ 1 and with −1 < yi < 1. The question of whether the function y = sin(ωx) can approximate these points arbitrarily closely for a suitable choice of ω is considered. It is shown that such approximation is possible if and only if the set {x1,..., xn} is linearly independent over the rationals. Furthermore, a constructive sufficient condition for such approximation is provided. The results provide a sort of counterpoint to the classical sampling theorem for bandlimited signals. They also provide a stronger statement than the well-known result that the collection of functions {sin(ωx) : ω < ∞} has infinite pseudo-dimension.