Fourier Series

Figure 1.3.1. Convergence of FS.Thus the series converges to the value of the function at points of continuity and to themidpoint between the two limits at a point of discontinuity. As it depends upon the limitsat a point of discontinuity, there is no requirement that the function is itself defined at thispoint.There are conditions that permit fx to be less well-behaved, while retaining convergenceof the series, but these are not needed at this point since the Dirichlet conditions areapplicable to most functions of interest. It is to be stressed that the Dirichlet conditions donot just provide a measure of convergence; they also provide a filtering mechanism todetermine whether or not a Fourier series representation can be obtained. For this reason,the function should be checked against the Dirichlet conditions before its Fourier series issought.Example 1Can the function defined by fx 1/4 − x 2 in −, be written as a Fourier series?No: It has infinite discontinuities at x 2 and so fails to satisfy the Dirichlet Conditions.Example 2Can the function defined by fx sin 1 in −, be written as a Fourier series?x−1No: The function has an infinite number of maxima and minima in the neighbourhood of19