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

AI Thread Summary
The discussion explores the relationship between the Fourier transform of a function and its inverse. It suggests that while a one-to-one function uniquely determines its inverse, there is no straightforward formula connecting their Fourier transforms. The participants agree that the transforms are related through this indirect relationship, but the interaction of the Fourier transform with function composition is complex. Ultimately, no simple or direct relationship exists between the Fourier transforms of a function and its inverse. The topic highlights the intricacies of Fourier analysis in relation to function inverses.
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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