Linear independence of time–frequency translates in Lp spaces

Abstract

We study the Heil-Ramanathan-Topiwala conjecture in L^p 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∈ L^p(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.