sin (ω x) Can Approximate Almost Every Finite Set of Samples

Consider a set of points (x_1,y_1),… ,(x_n,y_n) with distinct 0 ≤ x_i ≤ 1 and with -1 < y_i < 1 . The question of whether the function y = sin (ω x) 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 {x_1,… ,x_n} 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 (ω x) : ω < ∞} has infinite pseudo-dimension.