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) $