Schauder frames of discrete translates in Lᵖ(R)

We prove for every p > (1 + √5)/2 the following result, obtained earlier for p ≥ 2. There exist a uniformly discrete real sequence \λₙ\ₙ₌₁^∞ with |λₙ₊₁| > |λₙ|, a function g ∈ Lᵖ(R), and continuous linear functionals ^*ₙ\ on Lᵖ(R), such that every f ∈ Lᵖ(R) admits a series expansion \[ f(x) = ∑ₙ₌₁∞ g_n^*(f) g(x-λ_n) \] convergent in the Lᵖ(R) norm.