Relations Between Uniformly Almost Periodic Functions and the Fourier Transform

• Autor(a) principal: Araújo, Gonçalo Gomes
• Data de Publicação: 2022
• Tipo de documento: Dissertação
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: http://hdl.handle.net/10362/151950
• topic: Uniformly Almost Periodic Function; Fourier Transform; Banach Algebra; Inverse-Closed Algebra; Domínio/Área Científica::Ciências Naturais::Matemáticas

With this work, we intend to study the relation between uniformly almost periodic func- tions and the Fourier transform. To this end, we start by defining the concept of a uni- formly almost periodic function and we study several important algebraic and topological properties of these functions. Afterwards, we define a new class of functions, which we will call normal functions, and we will show that this class of functions is precisely equal to the set of uniformly almost periodic functions. We then define another class of functions, which we shall denote by AP (R), and we will define it as the closure, on L∞(R), of trigonometric polyno- mial functions, and we prove that this set also coincides with the set of uniformly almost periodic functions. We are then left with three equivalent definitions established. We then define the Fourier transform of a function belonging to L1(R) and, after studying some of its most important properties, we extend this concept to functions that belong to L2(R). After analyzing significant properties concerning Banach algebras, maximal ideals and multiplicative linear functionals, we define the algebra, APp(R) as the closure, in the norm of the Fourier multipliers, of trigonometric polynomial functions, and we conclude this paper by proving that the algebra APp(R) is inverse-closed in AP (R).