Algorithmic Approach for Formal Fourier Series

Abstract

The study of trigonometric series has started at the beginning of the nineteenth century. Joseph Fourier made the important observation that almost every function of a closed interval can be decomposed into the sum of sine and cosine functions. This technique to develop a function into a trigonometric series was published for the first time in 1822 by Joseph Fourier. The resulting series is nowadays called Fourier series. Since Fourier’s time, many different approaches to understand the concept of Fourier series have been discovered, each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on mathematical ideas and tools that were not available at the time Fourier completed his original work. Although the original motivation was to solve the heat equation for a metal plate, it later became obvious that the same technique could be applied to a wide variety of mathematical and physical problems and has many applications in electrical engineering, vibration analysis, acoustics, optics, signal treatment, image processing, etc. Despite the importance of Fourier series, the method used until now to compute them via computer algebra systems (CAS) is essentially based on the same principle as in Fourier’s time, i.e. by the evaluation of certain integrals. Unfortunately this technique is not completely successful for many functions. Although numeric values of the Fourier coefficients might be available, symbolic values are often not accessible. Modern CAS like Maple or Mathematica can compute such integrals in many cases for a given ${n \in \mathbb{Z}}$. However if one is interested in the Fourier coefficients for all ${n \in \mathbb{Z}}$, then n is considered as a given symbolic variable and such integrals can be computed only in few cases. In this paper we introduce an algorithmic approach to compute those Fourier coefficients, involving differential equations of a particular form, and recurrence equations. This approach extrapolates the computation of the Fourier series for functions for which the computation of Fourier coefficients via the definition is out of reach for current CAS.

A holonomic recurrence equation for a _n , i.e. a recurrence equation which is linear, homogeneous and has polynomial coefficients, can be written in operator notation as L(a _n ) = 0. The operator L can be interpreted as a non-commutative polynomial via the commutator rule Nn − nN = N, N denoting the shift operator Na _n  = a _n+1. In the last section we show how our algorithm can be used to factorize such recurrence operators in certain cases.