Extend uniformly discrete translate frames to all p > 1

Determine whether, for every p > 1, there exist a uniformly discrete sequence {λ_n} ⊂ R with |λ_n| = n + o(1), a function g ∈ L^p(R), and continuous linear functionals {g_n} on L^p(R), such that every f ∈ L^p(R) admits the L^p-convergent expansion f(x) = ∑_{n=1}^∞ g_n(f) g(x − λ_n).

Background

Theorem 1.1 of the paper establishes the existence of Schauder frames in L^p(R) formed by uniformly discrete translates for p ≥ (1+√5)/2, including quantitative control |λ_n| = n + o(1). The construction also admits a nonnegative generator (Theorem 5.1).

The authors highlight that the key remaining question is whether the same existence result extends to the full range p > 1, which would require overcoming technical barriers tied to the approximation lemma used in the proof.