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Given two functions $f$ and $g$, is there a formula for the Fourier transform of $f \circ g$ in terms of the Fourier transforms of $f$ and $g$ individually?

I know you can do this for the sum, the product and the convolution of two functions. But I haven't seen a formula for the composition of two functions.

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up vote 7 down vote accepted

There is no such rule in general. The key here is variable substitution: If $g$ is a bijection and smooth enough then, if all integrals exists: $$ (\widehat{f\circ g})(\xi) = \int f(g(x))\exp(ix\xi)dx = \int f(y)\exp(ig^{-1}(y)\xi)|\det g'(y)|^{-1}dy.$$ This does only rarely lead to something interesting, e.g. in the case of scaling (i.e. linear transformation of the variable): Working in $\mathbb{R}^d$ with $A\in\mathbb{R}^{d\times d}$ invertible: $$ (\widehat{f\circ A})(\xi) = |\det A^{-1}|\widehat{f}(A^{-T}\xi). $$

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