We discuss various properties of functionsfinLp(—π,π) whose Fourier coefficients vanish off a set. Two types of theorems are proved. The first gives certain implicit conditions onEso that certain functionsFwhich “majorize”falso belong toLp. The second deals with certain specific setsEwhich are lacunary on the negative integers. FunctionsfinLpwhose Fourier coefficients vanish offEbehave likeHpfunctions in several ways. Various open questions are raised.