Fourier transform over Lie group

let be the Lie Group of translations $y=x+a$ and dilations $y=bx$ whose generator are $\frac{d}{dx}$ and $x\frac{d}{dx}$

then could i define the Fourier transform over this group if i use a suitable measure ¿what should this measure be ? , how can i for example for these lie groups (or for other group if i know the generators) the Fourier integral ??

for example if i define the derivative by $D= \frac{d}{dx}$ then should the Kernel of the integral be $exp(-ixD)$

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I think what you came across is simply that the Fourier transform of the additive group of an locally compact algebra $A$ behaves well with respect to the scalling by invertible elements of $A$.

I would not call this the Fourier transform of the $a x +b$ group, but simply the Fourier transform of the $"b"$ part.

Note that the Fourier transform of an locally abelian group has a relation to the representation theory, where as your definition above can not be related to representation theory.

If you want an example of what comes close to a Fourier transform on a nonabelian Lie group, you can consider the Harish-Chandra transform and the Plancherel measure.

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Answer for let be the Lie Group of translations y=x+a and dilations y=bx whose generator are dx and dx/x

then could i define the Fourier transform over this group if i use a suitable measure ¿what should this measure be ? , how can i for example for these lie groups (or for other group if i know the generators) the Fourier integral ??

for example if i define the derivative by D= dx then should the Kernel of the integral be exp(−ixD)

Please go to google and just write On thee left ideals for the group algebra of the affine group by Kahar El Hussein , you will find this article and will be a good response on your questio

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