𝛼∼𝑙 relatively dense sets and weighted pseudo almost periodic phenomenon

Abstract

In this paper, we propose a new idea of studying the weighted pseudo almost periodic phenomenon. We introduce a new mathematical concept of α ∼ l \alpha \sim l relatively dense set and show that the α ∼ l \alpha \sim l relatively dense set is more useful than the classical relatively dense set due to Bohr or Bochner in treating the weighted pseudo almost periodic phenomenon. We prove the uniqueness of decomposition of the weighted pseudo almost periodic functions with values in Banach spaces and with weights in a new weight function set, and show that the new weight set is larger than all the existing weight function sets which ensures the uniqueness of such decomposition in literature. Moreover, we reveal several properties of the vector-valued weighted pseudo almost periodic functions, and provide a series of concrete examples and remarks for illustrating our theoretical results.