Linear independence of time-frequency translates

Abstract

Abstract. The refinement equation ϕ(t) = N 2 k=N 1 c k ϕ(2t − k) plays a key role in wavelet theory and in subdivision schemes in approximation theory. Viewed as an expression of linear dependence among the time-scale translates, it is natural to ask if there exist similar dependencies among the time-frequency translates e 2πibt f (t + a) of f ∈ L 2 (R). In other words, what is the effect of replacing the group representation of L 2 (R) induced by the affine group with the corresponding representation induced by the Heisenberg group? This paper proves that there are no nonzero solutions to lattice-type generalizations of the refinement equation to the Heisenberg group. Moreover, it is proved that for each arbitrary finite collectionis independent is an open, dense subset of L 2 (R). It is conjectured that this set is all of L 2 (R) \ {0}.

A survey on frame representations via dynamical sampling

“…The result in (i) was proved in [32] (hereby confirming a conjecture stated in [30]); (ii) & (iii) are classical results [27,12], and (iv) is proved in [5].…”

Frame properties of systems arising via iterated actions of operators

“…k=1 → H such that a given frame {f k } ∞ k=1 can be represented in the form {T n ϕ} ∞ n=0 for some ϕ ∈ H is independent of the ordering of the elements in {f k } ∞ k=1 . Proposition 2.3 has immediate consequences for Gabor systems in L 2 (R) : Example 2.4 It was conjectured in [15] and finally proved in [17] that for any a, b > 0 and any nonzero g ∈ L 2 (R) the Gabor system {E mb T na g} m,n∈Z is linearly independent. Proposition 2.3 therefore implies that there is a (possibly unbounded) operator T such that {E mb T na g} m,n∈Z = {T n ϕ} ∞ n=0 for some ϕ ∈ L 2 (R); one choice could be ϕ = g.…”

“…Note that [15] even conjectured that any Gabor system {E bm T am g} m∈I with g = 0 and distinct points {(a m , b m )} m∈I is linearly independent. Thus, if this is true, we would again obtain a representation {E bm T am g} m∈I = {T n ϕ} ∞ n=0 for some ϕ ∈ L 2 (R), under the assumption that I is countable and infinite.…”

HRT Conjecture and Linear Independence of Translates on the Heisenberg Group

“…The following conjecture known as the HRT conjecture [10,1,3,12,2,17,9] is an open problem deeply rooted in time-frequency analysis. It was posed about twenty years ago by Chris Heil, Jay Ramanathan, and Pankaj Topiwala in [11] as follows Conjecture 1. (The HRT Conjecture) Let φ ∈ L 2 (R) , φ = 0, and let F be a finite subset of R 2 .…”