We study the subspaces of $L_p(\mathbb{ R}^d)$ that consist of functions whose Fourier transforms vanish on a smooth surface of codimension $1$. We show that a subspace defined in such a manner coincides with the whole $L_p$ space for $p > \frac{ 2d}{ d+1}$. We also prove density of smooth functions in such spaces when $p < \frac{ 2d}{ d+1}$ for specific cases of surfaces and give an equivalent definition in terms of differential operators.
Categories: math.CA