The non-uniform convergence of the Fourier series for discontinuous functions is known as the Gibbs phenomenon. In 1899 American mathematician Josiah Willard Gibbs noticed that near a point where a function has a jump discontinuity, the partial sums of a Fourier series show a substantial overshoot near these endpoints. Carrying out the sums of the Fourier series to a higher number of terms will not diminish the amplitude of the overshoot although the overshoot occurs over a smaller and smaller interval. This overshoot exhibits itself in an oscillatory behavior near the discontinuous point(s) of the function. Although Wilbraham first analyzed this phenomenon in 1848, it was Gibbs that studied it in detail and for whom the behavior is named. Later, in 1906 Bôchner generalized this phenomenon to arbitrary functions. The Gibbs phenomenon is not only observed in Fourier series but also occurring at simple discontinuities in...