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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Analysis – On praise

November 4, 2013

Orders of infinity is Hardy’s monograph from 1910 on the work of Du Bois Reymond. From the preface:

There is, in Du Bois-Reymond’s original memoirs, a good deal that would not be accepted as conclusive by modern analysts. He is also at times exceedingly obscure; his work would beyond doubt have attracted much more attention had it not been for the somewhat repugnant garb in which he was unfortunately wont to clothe his most valuable ideas.