Are most Boolean functions determined by low frequencies?

We ask whether most Boolean functions are determined by their low frequencies. We show a partial result: for almost every function $f:\{-1,1{\}}^p\to \{-1,1\}$ there exists a function $f^{\mathrm{\prime }}:\{-1,1{\}}^p\to (-1,1)$ that has the same frequencies as $f$ up to dimension $(1\mathrm{/}2-o(1))p$.