What is so good about the $L^2$-norm of the second derivative being small?

Question

One of the main properties of cubic splines is the minimality property which basically means that if $s$ (cubic spline) and $g$ (some other function) interpolate $f$ in a certain way then $$\Vert s'' \Vert_2 \leq \Vert g'' \Vert_2.$$

First of all, I am somewhat surprised to encounter a stand-alone function under a norm, most of the time we care about a difference, e.g. $\Vert s - f \Vert$.

How do we interpret $\Vert s'' \Vert_2$? What is so good about the $L^2$-norm of the second derivative being small?

Answer 1

Let me elaborate on the comment of Exodd and provide some intuition behind the minimal property of splines.

I personally like to think about $\left\| s'' \right\|_2 $ as some kind of energy associated with function. A lot of people would call it the "energy functional", and below is my favorite analogy which explains why.

Consider a spaceship traveling in space. Assume $f(t)$ is the distance covered by the ship over time $t$. Then $f'$ is the speed function, and $f''$ is acceleration. By Newton's law of motion, the thrust of the spacecraft is proportional to the acceleration $f''$, therefore it is reasonable to say that the amount of "energy" (i.e. fuel) spaceship has spent over time $t$ is equivalent to $f''(t)$.

In such prospective the condition $\left\| s'' \right\|_2 \leq \left\| g'' \right\|_2$ means that $s$ interpolates $f$ with the least possible amount of "energy". As mentioned by Exodd, this feature allows to prevent redundant oscillation. Basically, the cubic spline will accelerate and decelerate the least among all the possible approximations of $f$ on a given interval. That means that it will jump up and down as little as possible, i.e. it will not oscillate unnecessarily (which makes sense, since the cubic spline is the polynomial of degree $\leq3$, so it has no more than $2$ local extrema).

If you would like to read something more formal on the topic, I would propose to start from this document, page 11.