MMP.12:5.3 - An accumulated quantity with an unstable derivative

In a continuous model on the normalized time interval [0,2*pi], accumulated activity is N(t), and its rate is r(t)=N’(t). Consider

N(t)=2*t; N_k(t)=2*t + sin(k*t)/k

for positive integers k. Their maximum difference is 1/k, tending to zero. Their rates are 2 and 2+cos(k*t), whose maximum difference remains 1. Both accumulated curves are nondecreasing. Thus nonnegativity of the rate does not remove this instability in the maximum norm.

A justified bound on rapid rate variation, or a penalty on changes in the rate, can suppress the oscillatory alternative. It also risks suppressing a real short surge. Specify which temporal detail the receiving use needs before selecting that structure.

If the use needs only the total over this interval, both curves give 4*pi. Recover that target from the endpoints without differentiating. For a fixed sampling interval a finite-difference inverse is continuous, but its error amplification grows as the interval shrinks; this differs from the discontinuity of the function-space inverse just exhibited.