Let Ω be an open subset of ℝⁿ. Denote by (formula presented) the closure in L^p(ℝⁿ) of the set of all functions ε ∈ L¹(ℝⁿ)∩L^p(ℝⁿ) whose Fourier transform has compact support contained in Ω. The subspaces of the form (formula presented) are called the spectral subspaces of L^p(ℝⁿ). It is easily seen that each spectral subspace is translation invariant; i.e., (formula presented) for all (formula presented). Sufficient conditions are given for the coincidence of (formula presented). In particular, an example of a set Ω is constructed such that the above spaces coincide for sufficiently small p but not for all p ∈ (0, 1). Moreover, the boundedness of the functional f → (Ff)(a) with a ∈ Ω, which is defined initially for sufficiently “good” functions in (formula presented), is investigated. In particular, estimates of the norm of this functional are obtained. Also, similar questions are considered for spectral subspaces of L^p(G), where G is a locally compact Abelian group.