Weierstrass CurveDate: 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/ |
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