Are the Fourier transforms of a function and its inverse related?

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SUMMARY

The discussion centers on the relationship between the Fourier transform of a function and the Fourier transform of its inverse function. It concludes that while a one-to-one function uniquely determines its inverse, there is no straightforward formula to relate the Fourier transforms of a function and its inverse. The Fourier transform does not interact with function composition in a simple manner, indicating that any potential relationship is indirect and complex.

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john1989
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Hi

Does anyone know if there is a relation between the Fourier transform of a function and the Fourier transform of the inverse function

in summary
FT[f(x)] ?= FT[f-1(x)]

Thanks!
 
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Logically they must be related because a 1-1 function uniquely determines its own inverse. Therefore, the Fourier transforms are at least related through this indirect relationship. However, I don't think there is any reasonable formula to relate the two. The Fourier transform does not interact with composition of functions in a simple way.
 

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