Consider the Fourier series
The th partial sum is given by
By integration by parts
uniformly for . Hence, if is fixed and , then , , or according as , , or ; compare (6.2.14).
These limits are not approached uniformly, however. The first maximum of for positive occurs at and equals ; compare Figure 6.3.2. Hence if and , then the limiting value of overshoots by approximately 18%. Similarly if , then the limiting value of undershoots by approximately 10%, and so on. Compare Figure 6.16.1.
This nonuniformity of convergence is an illustration of the Gibbs phenomenon. It occurs with Fourier-series expansions of all piecewise continuous functions. See Carslaw (1930) for additional graphs and information.