Question about function of matrices

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bumbar

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Can somebody help me with this problem
Does some function behaves like this:

\[f(D\cdot f(Dx))=f(D^2\cdot x)\]

Where \[D\] is a Matrix with elements \[d_{ij}\geq 0 \] ,
\[x\] is a vector with elements +1 or -1, and
\[f(\cdot)\] is a function

As you can see if \[f=y\] then the equation above is true.
But is there some other function then that one, for example does
\[sign\] makes the equation true, where \[sign(\cdot)\]
is the signum function
 

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