Schauder basis of translates in L^p(R)

Determine whether, for 1 < p < ∞, there exists a function g ∈ L^p(R) whose translates {g(· − λ_n)} form a Schauder basis of L^p(R). Equivalently, establish the existence (or nonexistence) of a Schauder basis in L^p(R) consisting solely of translates of a single function.

Background

The paper contrasts Schauder bases with Schauder frames and surveys known results about systems formed by translates. It is known that unconditional Schauder bases consisting of translates do not exist in L^p(R) for any 1 < p < ∞, and that uniform discreteness of the translation set is necessary for a basis of translates. However, the existence of (conditional) Schauder bases comprised solely of translates of a single generator remains a long-standing problem.

This problem is distinct from the Schauder frame results developed in the paper, which provide representations but need not be unique nor biorthogonal.