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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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up vote 2 down vote accepted

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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