On the connection between Sp-almost periodic functions defined on time scales and ℝ

Abstract

It is well known that a sufficient and necessary condition for a continuous function gg to be almost periodic on time scale T{\mathbb{T}} is the existence of an almost periodic function ff on R{\mathbb{R}} such that ff is an extension of gg. The purpose of this article is to extend these results to Sp{S}^{p}-almost periodic functions. We prove that the necessity is true, that is, an Sp{S}^{p}-almost periodic function on T{\mathbb{T}} can be extended to an Sp{S}^{p}-almost periodic function on R{\mathbb{R}}. However, a counterexample is given to show that the sufficiency is not true in general. By introducing a concept of minor translation set and characterizing the almost periodicity on T{\mathbb{T}} in terms of this new concept, we obtain a condition to ensure the sufficiency. Moreover, we show the necessity of this condition by a counterexample.

Keywords

• almost periodic
• sp-almost periodic
• time scales
• 42a75
• 26e70