Consider the Fourier series of the standard unit-amplitude square wave, which is an odd function of , as shown in Figure 2. The original function is given by the DP Fourier series
where , and the corresponding FC function is given by the DP Fourier series
where . These two series are convergent almost everywhere, but not absolutely or uniformly convergent. There are two special points at and at , where we have for the original function and . At these points the original function is discontinuous and the corresponding FC function diverges logarithmically. The representation of the original function in terms of the first-order center series is given by
where , for and , and the representation of the corresponding FC function in terms of the first-order center series is given by
where , for and . These two series are absolutely and uniformly convergent. The representation of the original function in terms of the second-order center series is given by
where , for and , and the representation of the corresponding FC function in terms of the second-order center series is given by
where , for and . These two series are absolutely and uniformly convergent. The derivation of the center series can be found in Section A.2 of Appendix A, and the results of the tests are shown in the tables in Subsections B.3 and B.4 of Appendix B.