Nonnegative Schauder bases in L^p(R) for p = 1 and p = 2

Determine whether L^p(R) admits a Schauder basis consisting entirely of nonnegative functions for p = 1 and p = 2.

Background

The literature has produced nonnegative Schauder bases in certain settings (e.g., L^1(0,∞) and L^2 spaces), and this paper constructs Schauder frames of translates with nonnegative generators. However, the existence of nonnegative Schauder bases specifically on L^1(R) and L^2(R) remains unresolved.

This question concerns the stronger demand of a basis (uniqueness of expansion) with the added structural constraint of pointwise nonnegativity of basis elements.