Linear independence of time–frequency translates in Lp spaces

Abstract

We study the Heil–Ramanathan–Topiwala conjecture in Lp spaces by reformulating it as a fixed point problem. This reformulation shows that a function with linearly dependent time–frequency translates has a very rigid structure, which is encoded in a family of linear operators. This is used to give an elementary proof that if f∈Lp(R), p∈[1,2], and Λ⊆R×R is contained in a lattice then the set of time frequency translates (f(a,b))(a,b)∈Λ is linearly independent. Our proof also works for the case 2 < p < ∞ if Λ is contained in a lattice of the form αZ×βZ.