(ωx) Can Approximate Almost Every Finite Set of Samples

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 (Formula presented.) 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 (Formula presented.)has infinite pseudo-dimension.