On Wikipedia, the Weierstass non-differentiable function is defined as: $$f(x)=\sum^{\infty}_{n=0}a^n\cos(b^n\pi x)$$

where $0<a<1$, $0<b$, and $ab>1+\frac 32 \pi$

Since it seems like, for any $a$ and $b$ that satisfy the above conditions, the resulting function will be continuous everywhere and differentiable nowhere, would it be more accurate to refer to the Weierstrass "function" as a family of functions? Are there a standard $a$ and $b$ that are used in discussing this function?


1 Answer 1


Are there a standard $a$ and $b$ that are used in discussing this function?

In my experientce, $a=1/2$ and $b=3$ are used most often, that is $$f(x)=\sum^{\infty}_{n=0}\frac{1}{2 ^n}\cos(3^n\pi x)$$ For example, these are the parameters are used on the illustration in Wikipedia article Weierstrass function.


Focusing on the more prominent local extrema on $[0,1]$, you will see repeated trisection of the interval, as in the construction of the standard Cantor set. This shows that $3^n$ was used inside the cosine. Also, the function evaluates to $2$ at $0$, implying $\sum_{n=0}^\infty a^n =2$, hence $a=1/2$.

Given that both trisection and scaling by $2^{-n}$ are very standard, I'll say that these are somehow canonical parameters.

But they don't satisfy $ab>1+\frac32 \pi$

Very true. That's an ugly constraint anyway, imposed to make the proof easier. Originally Weierstrass put $ab>1$, but then, apparently inspecting the details of the proof, changed to the above. Still, $ab>1$ is the natural condition: vertical scaling beats horizontal. Hardy eventually proved that even $ab\ge 1$ suffices for nowhere differentiability.

Recommended reading: Continuous nowhere differentiable functions, Master's Thesis by Johan Thim.

"Weierstrass functions" is fine

Just like Cantor sets are made using various scaling parameters, it makes sense to talk about Weierstrass functions.


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