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fourier analysis - Workarounds for a possible bug in the linearity of FourierTransform


This is somewhat similar to this question, except the problem I am encountering is to do with Fourier transforms of scalar multiples of functions and their derivatives.


I wish to input FourierTransform[a*f[t],t,x] and have Mathematica simplify it to a*FourierTransform[f[t],t,x], and, equivalently, to input FourierTransform[f'[t],t,x] and have Mathematica simplify it to ix*FourierTransform[f[t],t,x]. I'm taking the Fourier transform of a system of differential equations for functions f[t],g[t], etc., in order to instead only have to solve a system of algebraic equations for their Fourier transforms, but Mathematica seems to be having some problems doing this.


Note that LaplaceTransform works exactly as expected, but for some reason FourierTransform doesn't perform the expected simplification. If someone could suggest a solution that also incorporates the distributive property that FourierTransform was shown to have a problem with in the question I linked, that would be ideal.



This is essentially what I'd like FourierTransform to do:


LaplaceTransform[{a*f[t]+b*g'[t]==0,c*f'[t]+d*g[t]==0},t,x]


which returns:


{a*LaplaceTransform[f[t],t,x]+b*(-g[0]+x*LaplaceTransform[g[t],t,x])==0,
c*(-f[0]+x*LaplaceTransform[f[t],t,x])+d*LaplaceTransform[g[t],t,x]==0}

Instead, however, FourierTransform of the same expression returns:


{FourierTransform[a*f[t]+b*g'[t]==0,t,x],FourierTransform[c*f'[t]+d*g[t]==0,t,x]}


There is a solution to this problem here, although it doesn't actually explain FourierTransform's strange functionality.




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