ABSTRACT

This chapter summarises the basic theory of Fourier series. To understand the definition of Fourier series people will begin with the essential idea: representing a wave form in terms of frequency as opposed to time. The basic idea in Fourier series is to express a periodic wave as a sum of complex exponentials all of which have the same period. The chapter examines the validity of equating a periodic function with its Fourier series. The approach to the problem will be to examine the limit of partial sums of the Fourier series at any given point. The chapter discusses the following aspects of convergence of Fourier series: Gibbs' phenomenon, uniform convergence, and a more profound convergence theorem which applies to a wider class of functions that are encountered in measuring signals. Fourier sine series and cosine series are special forms of Fourier series for functions possessing either odd or even symmetry.