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.

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414/514 – The Schoenberg functions

January 23, 2012

Here is Jeremy Ryder’s project from last term, on the Schoenberg functions. Here we have a space-filling continuous map f:x\mapsto(\phi_s(x),\psi_s(x)) whose coordinate functions \phi_s and \psi_s are nowhere differentiable.

The proof that \phi_s,\psi_s are continuous uses the usual strategy, as the functions are given by a series to which Weierstrass M-test applies.

The proof that f is space filling is nice and short. The original argument can be downloaded here. A nice graph of the first few stages of the infinite fractal-like process that leads to the graph of f can be seen in page 49 of Thim’s master thesis.

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414/514 – Faber functions

January 17, 2012

Here is Shehzad Ahmed’s project from last term, on the Faber functions, a family of examples of continuous nowhere differentiable functions. Ahmed’s project centers on one of them, but the argument can be easily adapted to all the functions in the family.

As usual, the function is given as a series F=\sum_n f_n where the functions f_n are continuous, and we can find bounds M_n with \|f_n\|\le M_n and \sum_n M_n<+\infty. By the Weierstrass M-test, F is continuous. Also, the strategy for nowhere differentiability is typical: We associate with each point x a pair of sequences (a_n)_{n\ge0} and (b_n)_{n\ge0} with a_n strictly decreasing to x and b_n strictly increasing to x. The key lemma (shown, for example, in Johan Thim’s Master thesis available here) is that, if a continuous function f is differentiable at x, then we have

\displaystyle f'(x)=\lim_{n\to\infty}\frac{f(a_n)-f(b_n)}{a_n-b_n}.

In the case of the Faber functions, the functions f_n add `peaks’ in the neighborhood of any point, and the locations of these peaks can be used as the points a_n and b_n; moreover, the slopes accumulate at these peaks, and so the limit on the right hand side of the displayed equation above does not converge and, in fact, diverges to +\infty or -\infty.

Faber’s original paper, Einfaches Beispiel einer stetigen nirgends differentiierbaren Funktion, Jahresbericht der Deutschen Mathematiker-Vereinigung, (1892) 538-540, is nice to read as well. It is available through the wonderful GDZ, the Göttinger Digitalisierungszentrum.

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414/514 – Katsuura function

January 17, 2012

Here is Erron Kearns’s project from last term, on the Katsuura function, an example of a continuous nowhere differentiable function.

The presentation is nice: As usual with these functions, this one is defined as the limit of an iterative process, but the presentation makes it very clear the function is a uniform limit of continuous (piecewise linear) functions, and also provides us with a clear strategy to establish nowhere differentiability.

Actually, the function is presented in a similar spirit to many fractal constructions, where we start with a compact set K and some continuous transformations T_1,\dots,T_n. This provides us with a sequence of compact sets, where we set K_0=K and K_{m+1}=\bigcup_{i=1}^n T_i(K_m). Under reasonable conditions, there are several natural ways of making sense of the limit of this sequence, which is again a compact set, call it C, and satisfies C=\bigcup_{i=1}^n T_i(C), i.e., C is a fixed point of a natural “continuous” operation on compact sets.

This same idea is used here, to define the Katsuura function, and its fractal-like properties can then be seen as the reason why it is nowhere differentiable.

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414/514 – Continuous nowhere differentiable functions

October 22, 2011

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.

As a final project for the course, please choose an example of a continuous nowhere differentiable function, either from Thim’s thesis or elsewhere, and write a note on who it is due to and what the function is, together with complete proofs of continuity and nowhere differentiability. Feel free to add additional information you consider relevant for context.

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.

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.

The project is due Wednesday, December 14, by noon, but feel free (and encouraged) to turn it in earlier.

List of projects:

  • Diana Kruse: Bolzano function.
  • Jesse Tillotson: Weierstrass function.
  • Erron Kearns: Katsuura function.
  • David Sanchez: Peano function.
  • Shehzad Ahmed: Faber functions.
  • Chip Roth: McCarthy function.
  • Jeremy Ryder: Schoenberg functions.

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