$\sin (\omega x)$ Can Approximate Almost Every Finite Set of Samples

Abstract

Consider a set of points $(x_1,y_1),\ldots ,(x_n,y_n)$ with distinct $0 \le x_i \le 1$ and with $-1 < y_i < 1$. The question of whether the function $y = \sin (\omega x)$ can approximate these points arbitrarily closely for a suitable choice of $\omega$ is considered. It is shown that such approximation is possible if and only if the set $\{x_1,\ldots ,x_n\}$ 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 $\{\sin (\omega x) : \omega < \infty \}$ has infinite pseudo-dimension.