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• W4315851427 abstract "Abstract It is well known that a sufficient and necessary condition for a continuous function <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>g</m:mi> </m:math> g to be almost periodic on time scale <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi mathvariant=double-struck>T</m:mi> </m:math> {mathbb{T}} is the existence of an almost periodic function <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>f</m:mi> </m:math> f on <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi mathvariant=double-struck>R</m:mi> </m:math> {mathbb{R}} such that <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>f</m:mi> </m:math> f is an extension of <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>g</m:mi> </m:math> g . The purpose of this article is to extend these results to <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msup> <m:mrow> <m:mi>S</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msup> </m:math> {S}^{p} -almost periodic functions. We prove that the necessity is true, that is, an <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msup> <m:mrow> <m:mi>S</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msup> </m:math> {S}^{p} -almost periodic function on <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi mathvariant=double-struck>T</m:mi> </m:math> {mathbb{T}} can be extended to an <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msup> <m:mrow> <m:mi>S</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msup> </m:math> {S}^{p} -almost periodic function on <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi mathvariant=double-struck>R</m:mi> </m:math> {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 <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi mathvariant=double-struck>T</m:mi> </m:math> {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." @default.
• W4315851427 created "2023-01-13" @default.
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• W4315851427 date "2022-01-01" @default.
• W4315851427 modified "2023-09-26" @default.
• W4315851427 title "On the connection between <i>S</i> ^<i>p</i>-almost periodic functions defined on time scales and ℝ" @default.
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• W4315851427 doi "https://doi.org/10.1515/math-2022-0550" @default.
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