Weierstrass function

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.

sage1 sage2sage3The proof that f is continuous and nowhere differentiable can be found in Thim’s master’s thesis mentioned in the previous post. Weierstrass proof was published by DuBois-Reymond in 1875,

 Paul du Bois-Reymond. Versuch einer Classification der willkürlichen Functionen reeller Argumente nach ihren Aenderungen in den kleinsten Intervallen, J. Reine. Angew. Math., 79, (1875), 21–37.

Hardy’s improvement appeared in

Godfrey Harold Hardy. Weierstrass’s non-differentiable function, Trans. Amer. Math. Soc., 17 (3), (1916), 301–325. MR1501044.

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