Near a jump, a truncated Fourier series overshoots. Increasing the term count narrows the ringing region, but the limiting overshoot remains about 9% of the jump height. Pointwise convergence away from the jump does not imply uniform convergence.在跳跃间断点附近,截断的傅里叶级数会出现过冲。增加项数使振铃区域收窄,但极限过冲仍约为跳跃高度的 9%。远离间断点的逐点收敛,并不意味着一致收敛。
Three, eleven, and thirty-one terms are shown successively against a square wave. Watch the peaks beside zero rather than just the flatter sections.图中依次使用三项、十一项和三十一项,并以方波作参照。注意零点两侧的峰值,而不只是逐渐变平的区段。