A continuous, nowhere differentiable function: Part 1

[jbunniii]

jbunniii submitted a new PF Insights post

A Continuous, Nowhere Differentiable Function: Part 1

Continue reading the Original PF Insights Post.

 

[Greg Bernhardt]

Great first entry jbunniii! Looking forward to the rest of the series~

 

[mathman]

http://en.wikipedia.org/wiki/Weierstrass_function

Above describes classical description.

 

[vanhees71]

Great article! I remember that our analysis professor called this example the "Weierstraß Monster Function" :-)).

 

[Jimster41]

Love it. I have a book on power series that I got about 2 chapters into... Then got dropped like a bag of rocks. This rings a bell from that. Because power series are ineresting, and because I got dropped... About halfway down.

I want to understand... Z are the integers right. So the power series is on those. What does "Sup" mean, supremun (or max)? Also, and maybe this not something you can explain easily, where does the separating out of the real and imaginary parts come from again. I have a good cartoon of what Fourier's theorem says (from memories of a hellish "signals and Systems class) but operations with it quickly stupefy.

Kindof wish there was some pictures...

 

[Jimster41]

http://en.wikipedia.org/wiki/Weierstrass_function

Above describes classical description.

Ah, helpful.

 

[Jimster41]

Great article! I remember that our analysis professor called this example the "Weierstraß Monster Function" :-)).

Is that related at all to the crazy "Monstrous Moonlight" thingamajig? Is that one of the continuos but nowhere differentiable functions?

 

[Jimster41]

The necessary condition where the limit of the supremun of the Fourier coefficients = 1 can be interpreted as a discrete invariance to scaling? In other words is always looks the same periodically. Or is that wrong?

 

[jbunniii]

@Jimster41 - $\mathbb{Z}$ is the symbol for the set of integers. However, when working with power series we are generally summing complex numbers, and it is customary to use lower case $z$ to denote a general complex number. The reason it's useful to talk about power series in this context is because we have straightforward convergence theorems for them, and because a Fourier series of the form
$$\sum_{n=0}^{\infty} a_n e^{i n x}$$
is simply the power series
$$\sum_{n=0}^{\infty}a_n z^n$$
evaluated at $z = e^{ix}$, which is simply the unit circle since $|e^{ix}| = 1$.

"Sup" means supremum, which is the same as "max" when working with finite sets, but it also generalizes to infinite sets which may not have a maximum value. For example, the interval $[0,1)$, which is the set of all real numbers $x$ satisfying $0 \leq x < 1$, has no maximum value, but its supremum equals $1$.

"Lim sup" means "limit superior", and it is a notion associated with sequences of real numbers. It is the unique number $L$ such that only finitely many of the terms of the sequence exceed $L$, but infinitely many terms exceed any number smaller than $L$. So for example, if $a_n = (-1)^n$, the terms oscillate between $1$ and $-1$. The lim sup of this sequence is $1$. And the lim inf (which is defined analogously to lim sup) is $-1$. If the sequence has a limit, then the lim sup and lim inf are both equal to the limit.

Monstrous moonshine is related to the monster group from group theory, and is completely unrelated to this function. I haven't heard this function called the Weierstrass Monster Function until now, but I like it!

P.S. If you are reading this directly below the blog post and the typesetting comes out garbled, try reading it in the thread associated with the blog post: https://www.physicsforums.com/threa...entiable-function-part-1.811795/#post-5096693 It's exactly the same post, but for some reason it is not displaying correctly when viewed on the blog page.

 

[jbunniii]

P.S. I'm working on Part 2 today. That's where it gets really interesting.

 

[WWGD]

Is that related at all to the crazy "Monstrous Moonlight" thingamajig? Is that one of the continuos but nowhere differentiable functions?

If you are referring to monstrous moonshine, that refers to a simple group (group without non-trivial normal subgroups) of very large order.

 

[Jimster41]

@Jimster41 - $\mathbb{Z}$ is the symbol for the set of integers. However, when working with power series we are generally summing complex numbers, and it is customary to use lower case $z$ to denote a general complex number. The reason it's useful to talk about power series in this context is because we have straightforward convergence theorems for them, and because a Fourier series of the form
$$\sum_{n=0}^{\infty} a_n e^{i n x}$$
is simply the power series
$$\sum_{n=0}^{\infty}a_n z^n$$
evaluated at $z = e^{ix}$, which is simply the unit circle since $|e^{ix}| = 1$.

"Sup" means supremum, which is the same as "max" when working with finite sets, but it also generalizes to infinite sets which may not have a maximum value. For example, the interval $[0,1)$, which is the set of all real numbers $x$ satisfying $0 \leq x < 1$, has no maximum value, but its supremum equals $1$.

"Lim sup" means "limit superior", and it is a notion associated with sequences of real numbers. It is the unique number $L$ such that only finitely many of the terms of the sequence exceed $L$, but infinitely many terms exceed any number smaller than $L$. So for example, if $a_n = (-1)^n$, the terms oscillate between $1$ and $-1$. The lim sup of this sequence is $1$. And the lim inf (which is defined analogously to lim sup) is $-1$. If the sequence has a limit, then the lim sup and lim inf are both equal to the limit.

Monstrous moonshine is related to the monster group from group theory, and is completely unrelated to this function. I haven't heard this function called the Weierstrass Monster Function until now, but I like it!

P.S. If you are reading this directly below the blog post and the typesetting comes out garbled, try reading it in the thread associated with the blog post: https://www.physicsforums.com/threa...entiable-function-part-1.811795/#post-5096693 It's exactly the same post, but for some reason it is not displaying correctly when viewed on the blog page.

Very helpful. I can't believe I forgot that sum of e to complex powers version of the Fourier Theorem. Actually practically my all time favorite... I memorized it (slackly) as "You can make any continuous signal with a tuned sum of impulses, or the right collection of oscillators, which are like the same thing". Pretty shocking...

Yeah, I noticed the Tex doesn't seem to be working in the reply threads.

Dug out the book I have on "Fractals, Chaos, Power Laws" by Manfred Schroeder. I put it down last time because it just blew mind mind I think. Partly as I think about it because It finally explained why complex exponents are so... real.

I look forward to the second installment.

 

[Jimster41]

If you are referring to monstrous moonshine, that refers to a simple group (group without non-trivial normal subgroups) of very large order.

Yeah, that one. I've read the wiki on that at least twice now. I still have no idea what it means, except something like "Godzilla of primes" Maybe an insights for the future.