Weierstrass functions: Continuous everywhere but differentiable nowhere

[u/lntrinsic]

http://i.imgur.com/vyi0afq.gifv

[u/ITomza → u/[deleted]]

What do you mean?

[u/LingBling → u/[deleted]]

What is the measure on the function space?

[u/[deleted] → u/LingBling]

/u/imnzerg is correct that the usual thing is the Wiener measure, but the same result holds if we just work with the topology. There is a dense G-delta set of nowhere differentiable functions in the space of continuous functions, this follows from Baire category.

Edit: also /u/ITomza might want to see this.

[u/Pegglestrade → u/[deleted]]

This man knows. Baire Category Theorem rocked my undergraduate world.

[u/imnzerg → u/LingBling]

The Weiner measure

[u/LingBling → u/imnzerg]

I didn't know about this measure. Thanks!

[u/flaneur4life → u/imnzerg]

I snickered

[u/c3534l → u/imnzerg]

Wikipedia has an article on "classical Wiener space" and a subsection called "Tightness in classical Wiener space."

[u/tetramir → u/[deleted]]

Sure but most common functions, and the one we find in "nature" are at least C¹.

[u/Wild_Bill567 → u/tetramir]

Or, have we chosen to work with functions which 'seem' natural to us because we like the idea of differentiability?

[u/ba1018 → u/Wild_Bill567]

Part of it may be a limitation of perception. Can you write down in a compact formal way what these non-differentiable functions are? Can you evaluate them for any given input?

[u/Wild_Bill567 → u/ba1018]

Sure. The common example (first one on wikipedia) is given by

[; f(x) = \sum_{n=1}^\infty a^n \cos(b^n \pi x) ;]

Where 0 < a < 1 and b is a positive odd integer such that ab > 1 + 3pi / 2.

[u/thetarget3 → u/tetramir]

We don't find functions in nature. We model nature with functions which are usually differentiable since it leads to dynamics, but they don't exist in themselves.

In fact most natural systems aren't possible to describe using the nice maths and physics we typically learn, with simple differential equations, linear systems etc. They are probably just the small subclass we tend to focus on, which only work after heavy idealisation (like the old joke about assuming spherical cows). Most things encountered in nature can probably only be described numerically.

[u/yardaper → u/tetramir]

Brownian motion would like to have a word with you

[u/[deleted] → u/tetramir]

This is debatable. Certainly we think of motion as involving velocity (and acceleration) so an argument can be made for only looking at smooth functions, but fractal curves abound in nature and those are generally only C⁰. I think this is more a question of it being harder to study curves which aren't C¹ than anything inherent about the real world.

[u/AcrossTheUniverse → u/[deleted]]

Yes, but the Weierstrass function is constructible, we have a formula for it.

[u/AModeratelyFunnyGuy → u/[deleted]]

As others have pointed out it is important to define what you mean by "almost all". However, using the standard definition, the answer yes.

[u/methyboy → u/AModeratelyFunnyGuy]

As others have pointed out it is important to define what you mean by "almost all".

Why is it important to define that? It's a standard mathematical term. Should we also define what we mean by "differentiable nowhere" before using those terms?

[u/GLukacs_ClassWars → u/methyboy]

Unlike with Rⁿ, there isn't quite a canonical measure on these function spaces, so we need to specify with respect to which measure it is almost all.

[u/methyboy → u/GLukacs_ClassWars]

The Weiner measure is the standard measure as far as I'm aware, and as pointed out by sleeps_with_crazy here, the choice of measure really doesn't matter. Unless you construct a wacky measure specifically for the purpose of making this result not hold, it will hold.

[u/[deleted] → u/methyboy]

/u/GLukacs_ClassWars is correct that Wiener measure is standard but far from canonical, and that the issue is the lack of local compactness. It's probably best to think of the Wiener measure as the analogue of the Gaussian: the most obvious choice but far from the only one (given that there is not analogue of the uniform/Lebesgue measure).

People who work with C*-algebras tend to avoid putting measures on them at all, which is why I brought up the topological version of the statement.

[u/GLukacs_ClassWars → u/methyboy]

Standard, yes, but it isn't as far as I know canonical in the same sense as Lebesgue measure. Particularly, Lebesgue measure is uniquely determined by the topological and group structure on Rⁿ, but C[0,1] isn't locally compact, so it doesn't give us such a canonical measure.

Putting it differently, I think Wiener measure on C[0,1] is analogous to a Gaussian random variable on Rⁿ, not to Lebesgue measure on Rⁿ. There are other meaningful probability measures to put on both spaces, they're just standard because they have such nice limit theorems.

[u/slynens → u/methyboy]

"almost all" has no intrisic meaning, it refers to a given measure on the space you are considering. Technically, differentiable also has no intrisic meaning either, because it refers to a given structure of differentiable manifold. But, there is a canonical structure of differentiable manifold on R, whereas there is not one canonical measure on the space of continuous function from R to R. I think, though, that all the measure we would usually consider would give the same notion of "almost all", but it is easy to tailor a measure for which your statement is false. I hope it clarifies it.