Weierstrass Curve

Date: 09/13/2002 at 01:11:24
From: Colin
Subject: Weierstrass curve

I read in an old article written by Alfred Adler that about 100 years 
ago the mathematician Weierstrass gave an example of a curve 
consisting of angles, or corners, and nothing else. Where can I find 
this equation?

Date: 09/13/2002 at 10:26:59
From: Doctor Fenton
Subject: Re: Weierstrass curve

Hi Colin,

Weierstrass's original example was a trigonometric series, and 
checking the literature, I found several different versions of it, so 
I'm not sure exactly which version was Weierstrass's.

One version (Gelbaum and Olmstead, _Counterexamples in Analysis_) is

   oo
   ---
   \    b^n cos(a^n pi x)
   /
   ---
   n=0

where b is an odd integer, 0<a<1, and ab > 1+(3*pi/2).

T. W. Körner, in _Fourier Analysis_, gives the example as

   oo
   ---
   \    a^(-n) sin(b^n x)
   /
   ---
   n=-oo

where b is an integer, and b/a and a are "sufficiently large."

Neither book proves the properties of the function; it appears to be
pretty complicated to analyze.  Instead, many books prove the result
for similar but simpler functions.

Walter Rudin uses a function g(x)=|x| for |x| < 1, and extends g(x) to 
the real line by making it periodic of period 2.  He proves that

    oo
   ---
   \    (3/4)^n g(4^n x)
   /
   ---
   n=0  

is continuous and nowhere differentiable. Michael Spivak has a similar 
example in his text _Calculus_.

Körner also proves that

     oo
   ---
   \    (n!)^(-1) sin((n!)^2 x)
   /
   ---
   n=0 

is continuous and nowhere differentiable.

I did a quick Web search using keywords "Weierstrass" and 
"nondifferentiable" and found many references, but no explicit
formulas, but then I didn't check many sites. You might also use
"nowhere differentiable" instead of nondifferentiable if you want
to try that search.

If you have any questions, please write back.

- Doctor Fenton, The Math Forum
  http://mathforum.org/dr.math/