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Consider a set of functions $f:[0,1]\rightarrow \mathbb{R}$ with $f(0) = f(1) = 0$ that are supposed to approximate the function $\mathbf{0}:[0,1]\rightarrow \mathbb{R}$ with $\mathbf{0}(x) \equiv 0$. The graph of $\mathbf{0}(x)$ is to be seen as the “reference curve”.

Since the graph of $\mathbf{0}$ is the shortest curve between its end points, it seems natural to say that $f$ approximates $\mathbf{0}$ better than $g$ when the arc length of (the graph of) $f$ is shorter than that of $g$, i.e.

$\int_0^1\sqrt{1 + f'(x)^2}dx < \int_0^1\sqrt{1 + g'(x)^2}dx\quad\quad\quad (1_0)$

But this conflicts with another measure of proximity: one can say $f$ approximates $\mathbf{0}$ better than $g$ when it deviates less from $\mathbf{0}$ than $g$ in the following sense

$\int_0^1 |f(x) - \mathbf{0}(x)| dx < \int_0^1 |g(x) - \mathbf{0}(x)| dx\quad\quad\quad (2_0)$

which means

$\int_0^1 |f(x)| dx < \int_0^1 |g(x)| dx\quad\quad\quad (2_1)$

which holds – for the sake of comparison with $(1)$ – iff

$\int_0^1\sqrt{1 + f(x)^2}dx < \int_0^1\sqrt{1 + g(x)^2}dx\quad\quad\quad (2_2)$

The “conflict” can easily be made visible:

enter image description here

The red curve $g$ has the same length as the blue one $f$ while the area under $g$ is obviously greater than the area under $f$.

There is another – “intrinsic” – property of the blue curve $f$ that makes it a better approximation than the red curve $g$ – and even without an explicit reference to $\mathbf{0}$: it changes its direction “more often”. In integral notation:

$\int_0^1 |f''(x)| dx > \int_0^1 |g''(x)| dx\quad\quad\quad (3_0)$

or equivalently:

$\int_0^1\sqrt{1 + f''(x)^2}dx > \int_0^1\sqrt{1 + g''(x)^2}dx\quad\quad\quad (3_1)$

In which “theory” or framework do these findings fit together – and how?

And what are, resp., the official names of

$\int |f(x)| dx $

$\int |f'(x)| dx $

$\int |f''(x)| dx $

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It depends on what you want to do. If you are interested in approximating $0$ by functions $f$ in the sense that $f$ is similar to $0$ and has similar length, then you could measure that with the integrals in $(1_0)$ and $(2_2)$. Only controlling the length would not suffice, since your approximating curves could then lie very far apart from $0$, though I think they would have to converge to some constant function eventually.

However, I don't see how changing the direction makes a better approximating for a straight line.

The integrals of the absolute values of the derivatives of $f$ could be called $L^1$-norms of $f$ and its derivatives. You might be interested in looking up »Sobolev spaces«.