I've started working with Holder spaces recently and I'm wondering how I should think of them intuitively? I really have no idea what a function $f$ that is Holder continuous with exponent $\alpha$ is supposed to look like whereas I do have a good idea for other function spaces.
$L^{\infty}$:
Say we had a function $f$ such that $\Vert f \Vert_{L^\infty([0, 2])} \le 1$. Then I can visualize $f$ as being some function in the box $[0, 2] \times [-1, 1]$.
Lipschitz: If we take another function $g$ and say it is Lipschitz continuous with some fixed constant $C$ I know that the slope of the $g$ will always be less than $C$ at any point in its domain which agrees with the vizualization stated on the Wikipedia page regarding a double cone that can be translated along the graph such that the graph always remains outside the cone.
Holder:
What is the best way to visualize a Holder continuous function with exponent $\alpha$ on a given domain such as, say, $[0, 5]$? Does such a function have a clear geometric interpretation like $L^{\infty}$ functions? What would be an example of such a function if the exponent was $\alpha = 0.2$ for example? Would a function with an exponent of $\alpha = 0.3$ be 'nicer' than one with $\alpha= 0.2$?
Best Answer
Hoelder continuity is about the roughness of a path. So there are some extremes. First of all if $f$ is $\alpha$ Hoelder continuous with $\alpha>1$, then $f$ is constant (very easy to prove).
A function that is Hoelder continuous with $\alpha=1$ is differentiable a.e.
So if you're Hoelder continuous with $\alpha\ge 1$ things are very nice. Less than $1$ and things are much less nice.
The lower your Hoelder exponent is, the rougher the path is. In particular $\alpha=\frac12$ is very critical. I do research in rough path theory, which handles the case when $\alpha<\frac12$ but this is much more advanced.
I think the best way to understand different Hoelder continuity is to look at some paths! I do work with what's called fractional Brownian motion. Fractional Brownian motion is a stochastic process that has a parameter $H$. $H$ ends up being the Hoelder exponent of the path a.s.
So look at some fractional Brownian motions! Here's a few pictures from Wikipedia that I think will really help clarify how "rough" a path is.
Also to answer your question, would a Hoelder continuous path with $\alpha=.3$ be nicer than $.2$, YES.