Positive orthogonal functions

TL;DR

This paper resolves an open question from the 1980s regarding the existence of positive orthogonal functions in subspaces of almost periodic functions, specifically for subspaces defined by three periods, with implications for oscillatory behavior analysis.

Contribution

It provides a complete clarification of the existence of positive orthogonal functions for subspaces with three periods, answering a long-standing open problem.

Findings

Resolved the open question on positive orthogonal functions for three-period subspaces.

Clarified the conditions for existence and non-existence of such functions.

Extended the understanding of oscillatory behavior in almost periodic functions.

Abstract

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.