Fourier transform of an an (almost) periodic function

Question

I am not sure about the terminology here but I have a function for which I know is almost periodic in the sense that for integer values of $k$ we have

$$f(x+k)=e^{-|k|}f(x)$$

Is there any way to find the Fourier transform of such a function?

Answer 1

The only function verifying this is the function $0$. If you take the Fourier transform with respect to $x$ on both sides you find indeed $$ e^{2i\pi kx}\,\widehat{f(x)} = e^{-|k|}\,\widehat{f(x)} $$ and so taking absolute values $$ |\widehat{f(x)}| = e^{-|k|}\,|\widehat{f(x)}| $$ and the only solution when $k\neq 0$ is $f=0$.

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Notice that it can be proved without using the Fourier transform by considering your equation with $k=1$ at the point $x$ and $k=-1$ at the point $x+1$, leading to $f(x) = e^{-1} \, f(x+1)$ and $f(x+1) = e^{-1} \, f(x)$. Thus $$ f(x) = e^{-2} \, f(x) $$ or equivalently, $f=0$.