Functions whose Fourier transform vanishes on a surface

Abstract

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.

11 pages. Theorems 3 and 4 of this preprint follow by duality from more general Theorem 1 in "$L^p$-integrability, supports of Fourier transforms and uniqueness for convolution equations" by M. L. Agranovsky and E. K. Narayanan in Journ. Four. Anal. Appl. vol.10:3 (2004), 315--324

References in corpus (1)

• The Helmholtz equation with $L^p$ data and Bochner-Riesz multipliers

Cited by in corpus (1)

• Sets of $p$-restriction and $p$-spectral synthesis