Fourier partial sums of a function with a jump overshoot near the discontinuity by about 9% of the jump — and that overshoot never goes away, it only narrows. Pick a 1-periodic function and a way to sum its Fourier series: the sharp Dirichlet cutoff shows the overshoot, while Fejér and Bochner–Riesz smoothing tame it. The kernel panel shows why: Fejér’s kernel is non-negative, Dirichlet’s is not.
An original interactive applet, part of tao-web.
With real coefficients f(x) = a0/2 + ∑ [an cos 2πnx + bn sin 2πnx], each summation method multiplies the n-th term by m(n/N): Dirichlet by 1 (sharp cutoff), Fejér by 1 − |t|, Bochner–Riesz by (1 − t2)δ (so δ = 0 is Dirichlet again). The Dirac comb has every coefficient equal to 1, so its partial sum is literally the kernel KN. The overshoot comes from the kernel’s negative lobes; the triangle wave is continuous, so its series converges uniformly with no overshoot at all.
Related: the Kakeya and other tao-web applets.