Infinite product $\prod_{ m = 1 }^{ \infty } { \cos (\frac{ x }{ 4 ^ { m }}) }$ I'm trying to figure out this infinite product: $
   f  (  x )   =  \prod_{ m = 1 }^{  \infty  } {  \cos (\frac{ x }{ 4  ^ { m }}) } $
I did the same trick as in the $2^m$ version to get this functional equation: $
 f(x)  \times  f(2x)  = \frac{ \sin ( x )}{ x }
$
And now im really stuck, I'm not even sure if there's only one solution for this equation. I thought convolution might work but that was messy..
My main question is how to solve this infinite product, but anything about this equation will be great too.
Thanks!
What i did was:
$
 f(x)=  \prod_{ m = 1 }^{  \infty  } {  \cos (\frac{ x }{ 4  ^ { m }}) }  =  \lim_{N \to  \infty }   \prod_{ m = 1 }^{ N } {  \cos (\frac{ x }{ 4  ^ { m }}) }    =  \lim_{N \to  \infty }   \prod_{ m = 1 }^{ N } { \frac{ \sin (\frac{ x }{ 4  ^ {   m  -  1   }})}{   4  \times  \sin (\frac{ x }{ 4  ^ { m }})   \times  \cos (\frac{ 2x }{ 4  ^ { m }})} }    =  \lim_{N \to  \infty }  \frac{ \sin ( x )}{   4  ^ { N } \times  \sin (\frac{ x }{ 4  ^ { N }})   \times  \prod_{ m = 1 }^{ N } {  \cos (\frac{ 2x }{ 4  ^ { m }}) } }   = \frac{ \sin ( x )}{ x }   \lim_{N \to  \infty }  \frac{ x }{ 4  ^ { N } \times  \sin (\frac{ x }{ 4  ^ { N }})}     \lim_{N \to  \infty }  \frac{ 1 }{ \prod_{ m = 1 }^{ N } {  \cos (\frac{ 2x }{ 4  ^ { m }}) } }   = \frac{ \sin ( x )}{ x } \times  1  \times \frac{ 1 }{ f    \left(  2x  \right) } $
 A: Comments.  Probabilistic viewpoint:
Let $X_n, n=1,2,3,\dots$ be IID random variables with
$\mathbb P(X_n = 1)=\mathbb P(X_n = -1)=\frac12$.  Fix $r \in (0,1)$.Consider the rancom variable
$$
Z_r = \sum_{n=1}^\infty r^n X_n .
$$
Then your function is $f_{1/4}$, where
$$
f_r  (  x )   =  \prod_{ m = 1 }^{  \infty  }   \cos \left(r^n x\right) 
$$
is the characteristic function of $Z_{r}$, that is:
$$
\mathbb E\left[\exp\big(ixZ_{r}\big)\right] = f_r(x) .
$$
In the case $r=1/2$, note that $Z_{1/2}$ is uniformly distributed on $[-1,1]$, so
$$
f_{1/2}(x) = \frac12\int_{-1}^1 e^{ixz}\,dz = \frac{\sin x}{x} .
$$
But for the case in this question, $r=1/4$, the random variable $Z_{1/4}$ has a singular distribution, concentrated on a Cantor set.  (A fractal measure.)  And $f_{1/4}$ is the characteristic function of that fractal measure.

But of course "known" functions do not have continuous singular Fourier transforms; so it is unlikely that $f_{1/4}$ can be written in terms of standard special functions.
