A Transformada de Fourier: da motivação à equação do calor numa barra infinita

Abstract

This paper presents an overview of the Fourier Transform, its properties and applications, with a particular emphasis on the space of rapidly decreasing functions, which is known as the Schwartz space. The Fourier transform is a fundamental tool in mathematical analysis, employed for the resolution of partial differential equations. This enables differential equations to be converted into more readily manageable algebraic equations. The methodology adopted is based on an analysis of the Schwartz space and its properties, which are essential to ensure the proper behavior of functions in the context of the Fourier transform. Subsequently, the principal properties of the Fourier Transform are examined, including its linearity, differentiability and applicability within the context of Schwartz space, as well as its inverse transform. This work was developed based on an investigation of bibliographical references and theoretical materials listed at the end of this paper. The results obtained demonstrate the significance of the Fourier Transform in determining the solution to the heat equation in an infinite bar, thereby facilitating the identification of a potential solution for the associated partial differential equation. It is our intention that this work provides a clear and comprehensive overview of the Fourier Transform, its properties and its theoretical applications, thus establishing it as an essential tool in analysis.