.† The American mathematician Josiah Willard Gibbs

Chapter 10, Problem 39E

(choose chapter or problem)

PROBLEM 39EGibbs Phenomenon.† The American mathematician Josiah Willard Gibbs (1839–1903) observed that near points of discontinuity of f, the partial sums of the Fourier series for f may overshoot by approximately 9% of the jump, regardless of the number of terms. This is illustrated in Figure 10.12, on page 594, for the function whose Fourier series has the partial sums To verify this f(x), for proceed as follows:(a) Show that (b) Infer from part (a) and the figure that the maximum occurs at and has the value (c) Show that if one approximates (d) Use the result of part (c) to show that the overshoot satisfies (e) Using the result of part (d) and a numerical integration algorithm (such as Simpson’s rule, Appendix C) for the sine integral function show that Thus, the approximations overshoot the true value of of the jump from Figure 10.12