Fórmulas de cuadratura. Aplicación ao cálculo de coeficientes de Fourier

Abstract

[GL] Neste traballo presentaremos diferentes fórmulas de cuadratura, centrándonos na fórmula do trapecio composta e con especial interese nun tipo de converxencia que esta presenta, chamada converxencia superalxébrica, onde se mellora a orde 2 de converxencia. Tras iso, introducirémonos nas series de Fourier, veremos como obter os coeficientes da citada serie empregando fórmulas de cuadratura e abordaremos algúns resultados en canto á converxencia. Máis tarde, pasaremos a estudar a transformada discreta de Fourier (DFT) e un algoritmo para calculala de maneira rápida e eficaz, chamado transformada rápida de Fourier (FFT). Finalmente, implementaremos un método que fará uso da FFT para calcular os coeficientes de Fourier mediante a fórmula de cuadratura do trapecio composta, rematando con varios casos prácticos nos que daremos fe da eficacia do método.

[EN] In this work we will present different quadrature rules, focusing on the composite trapezoidal rule and paying special attention to the superalgebraic convergence, a type of convergence where the order 2 of convergence is improved. After this, we will introduce the Fourier series, we will see how to obtain the coefficients of such series using quadrature formulas and we will discuss some results about convergence. Afterwards, we will study the discrete Fourier transform (DFT) and an algorithm to compute it quickly and efficiently, called the fast Fourier transform (FFT). Finally, we will also implement a method that will make use of the FFT to calculate the Fourier coefficients by means of the composite trapezoidal rule, ending with several practical cases in which we will demostrate the effectiveness of the algorithm.