Positive orthogonal functions

The existence or non-existence of positive orthogonal functions for subspaces of almost periodic functions has important applications in studying the oscillatory behavior of vibrations. Cazenave, Haraux and Komornik have obtained a number of theorems of this type. The purpose of this paper is to answer an open question formulated in the 1980's, and to completely clarify the situation for subspaces defined by three periods.