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Let $\mathcal{F[\omega]}$ denote the space of functions whose Fourier transforms are supported in $[-\omega,\omega]$.

  • Is $\mathcal{F[\omega]}$ closed under composition? If not, what is its closure?

  • What about $\bigcup_{0<\omega<\infty} \mathcal{F[\omega]}$?

  • Do the answers generalize to maps between arbitrary-dimensional Euclidean spaces?

Edit: I realized that for $\mathcal{F[\omega]}$ to have a chance of being closed under composition, the functions' Fourier transforms should be bounded in amplitude somehow. I guess that bounding the $\mathbf{L}^2$ norm (of the transform) should be enough, but I'm not yet sure...

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    $\begingroup$ Perhaps consider $\operatorname{sinc}(x) = \sin(\pi x) / (\pi x) \in \mathcal{F}[\pi]$, but the Fourier transform of $\operatorname{sinc}\circ \operatorname{sinc}$ might be hard to find (I tried and failed with Wolfram Alpha). For polynomials $p$, the highest frequency in $p \circ g$ is related to the highest frequency of $g$ and the degree of $p$, but $\operatorname{sinc}$ is not a polynomial... $\endgroup$
    – Claude
    Mar 29, 2018 at 16:38
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    $\begingroup$ In which space are you are considering the Fourier transform? If it is the space of tempered distributions or $L^2$ then the answers to your questions follow from the Paley-Wiener theorem and they are: no, yes and yes. $\endgroup$
    – pipenauss
    Mar 29, 2018 at 22:44
  • $\begingroup$ @pipenauss, thanks for this. I'm unfamiliar with Fourier analysis and have only just encountered the Paley-Wiener theorems; if you could outline how to derive these conclusions from them, I would be very grateful. $\endgroup$ Mar 29, 2018 at 23:55
  • $\begingroup$ I can see from the statement of the theorem that, at least for $\omega$ fixed, there are going to be some functions in $\mathcal{F}[\omega]$ (as originally defined above) whose composition is not in $\mathcal{F}[\omega]$. But I have no idea whether that's just for the reason I stated in the edit above - intuitively, that if the derivative of $g$ is large enough for long enough then "the chain rule" means that $g \circ f$ can contain higher frequencies than $f$ - or whether there's something more fundamental going on, that will cause problems even if we impose further restrictions. $\endgroup$ Mar 30, 2018 at 1:07
  • $\begingroup$ $f$ and $g$ should have been the other way round in the previous comment. So my question now is: let $\mathcal{F}_{lin}[\omega]$ denote the space of functions whose Fourier transforms are supported in $[-\omega,\omega]$ and whose derivatives are everywhere $\le 1$ in absolute value. Is $\mathcal{F}_{lin}[\omega]$ closed under composition? If so, is there an even broader subset of $\mathcal{F}[\omega]$ which is too? If not, is there a narrower subset (still not contained in $\mathcal{F}[\omega']$ for $\omega' < \omega$) which is? Should I post a separate question or just update the current one? $\endgroup$ Mar 30, 2018 at 15:35

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