Consider the function $f(x)=\sin( \log x)$ defined over $x>0$.

It has the cool feature that when you plot it, and change the x scale, it's overall shape does not change much. For example if you look at it over the $x$ range $[0,\;0.001]$ or $[0,\;1000]$ it's overall shape doesn't change.

Here is the question: Does there exist any positive real number $c$ that:

$f(c)=f(1)$ and $f(2c)=f(2)$ and $f(3c)=f(3)$ simultaneously?

Can we build a class of functions like $\sin ( \log x)$ that can form a base similar to fourier?

  • $\begingroup$ Could you please elaborate on the meaning of the last question? What would meet the criteria of being "a class of functions such as $\sin(\log(x))$"? $\endgroup$ Apr 8, 2011 at 3:26
  • $\begingroup$ @svenkatr: I think he means a real number $c$ for which $\sin(\log(c))=\sin(\log(1))$, $\sin(\log(2c))=\sin(\log(2))$, and $\sin(\log(3c)) = \sin(\log(3))$. $\endgroup$ Apr 8, 2011 at 3:32
  • $\begingroup$ @Jonas: I think he means a set of such "similar" functions such that any periodic function can be approximated arbitrarily well by a linear combination of functions in the set. As for what counts as "similar", I imagine only the OP knows. $\endgroup$ Apr 8, 2011 at 3:33
  • 2
    $\begingroup$ @Alex: That is why I asked the OP :) @Arturo: $c=e^{2\pi}$ meets those criteria. As would $e^{2\pi\cdot n}$ for any integer $n$. $\endgroup$ Apr 8, 2011 at 3:33
  • $\begingroup$ @svenkatr: That comment should be an answer. $\endgroup$
    – user856
    Apr 8, 2011 at 3:38

2 Answers 2


Consider $c = e^{2 \pi}$. Then, $f(nc) = f(n)$ for all $n$.

EDIT: I had put this as a comment at first, but I made it an answer on Rahul Narain's suggestion.


As for the second question, you could consider series in sin(n log x) and cos(n log x) for integers n, which correspond to Fourier series after the change of variable log x = t.

  • $\begingroup$ Can you explain this a little more ? for example, can we expand $Sin(x)=\sum{a_n Sin(n log(x)+b_n Cos(n log(x))}$ ? $\endgroup$
    – Austin
    Apr 8, 2011 at 10:59
  • $\begingroup$ I added a separate question for that: link $\endgroup$
    – Austin
    Apr 8, 2011 at 11:40

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