Weierstrass function

November 7, 2013

Weierstrass function from 1872 is the function f=f_{a,b} defined by

\displaystyle f(x)=\sum_{n=0}^\infty a^n\cos(b^n\pi x).

Weierstrass showed that if

  • 0<a<1,
  • b is an odd positive integer, and
  • \displaystyle ab>1+\frac32\pi,

then f is a continuous nowhere differentiable function. Hardy proved in 1916 that one can relax the conditions on a,b to

  • 0<a<1,
  • b>1, and
  • ab\ge 1.

Here, I just want to show some graphs, hopefully providing some intuition to help understand why we expect f to be non-differentiable. The idea is that the cosine terms ensure that the partial sums  \displaystyle f(m,x)=\sum_{n=0}^m a^n\cos(b^n\pi x), though smooth, have more and more “turns” on each interval as m increases, so that in the limit, f has “peaks” everywhere. Below is an animation (produced using Sage) comparing the graphs of f(m,x) for 0\le m<20 (and -10\le x\le 10), for a=1/2 and b=11, showing how the bends accumulate. (If the animations are not running, clicking on them solves the problem. As far as I can see, they do not work on mobiles.)

sage0Below the fold, we show the same animation, zoomed in around 0 by factors of 100, 10^4, and 10^6, respectively, illustrating the fractal nature of f.

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Continuous nowhere differentiable functions

November 7, 2013

Following a theme from two years ago, we will have a final project for this course, due Wednesday, December 18, by noon, but feel free (and encouraged) to turn it in earlier. (As discussed in lecture, the project is voluntary for some of you. Contact me if you are not sure whether it is required or voluntary for you.)

There are many excellent sources on the topic of continuous nowhere differentiable functions. Johan Thim’s Master thesis, written under the supervision of Lech Maligranda, is available online, here, but feel free to use any other sources you find relevant.

Please choose an example of a continuous nowhere differentiable function, either from Thim’s thesis or elsewhere, and write (better yet, type) a note on who it is due to and what the function is, together with complete proofs of continuity and nowhere differentiability. Though not required, feel free (and encouraged) to add additional information you consider relevant for context.

(For an example of what I mean by relevant additional information: Weierstrass function is \displaystyle f(x)=\sum_{n=0}^\infty a^n\cos(b^n\pi x) where 0<a<1, b is an odd positive integer, and \displaystyle ab>1+\frac32\pi. It may be interesting to add a discussion of precisely what conditions are needed from a,b to ensure (continuity and) nowhere differentiability; Weierstrass original requirements are more restrictive than necessary. For another example, Schoenberg functions, discussed in Thim’s thesis, give a natural example of a space filling curve, so consider including a proof of this fact.)

Please take this project very seriously (in particular, do not copy details from books or papers, I want to see your own version of the details as you work through the arguments). Feel free to ask for feedback as you work on it; in fact, asking for feedback is a good idea. Do not wait until the last minute. At the end, it would be nice to make at least some of the notes available online, please let me know when you turn it in whether you grant me permission to host your note on this blog.

Here is a list of the projects I posted on the blog, from last time:

Contact me (by email) as soon as you have chosen the example you will work on, to avoid repetitions; I will add your name and the chosen example to the list below as I hear from you.

List of projects:

  • Joe Busick: Katsuura function.
  • Paul Carnig: Darboux function.
  • Joshua Meier: A variant of Koch’s snowflake.
  • Paul Plummer: Lynch function.
  • Veronica Schmidt: McCarthy function.