When a function $f(t)=exp(-|t|)$ for example undergoes Fourier Transformation, it gives $F(w)=\frac{-2}{1+w^2}$
But what happens to the result if the time scale is scaled and shifted, so that $t \rightarrow\ t^* =at+b $ ?
How will the Fourier Transformation of the function change?
Edit: Following is the approach I took but is unsure about it's correctness
$Since \ t \rightarrow\ t^* =at+b \\ f(t) \rightarrow\ f(at+b) = e^{-|at+b|}) \\ therefore \\ F(w) = \frac{e^{-iwb}}{|-a|} \ * \frac{2}{1+(\frac{w}{-a})^2} $
The part I'm most uncertain about is $e^{-|at+b|}$ where there is the absolute value of at+b. I'm only treating it as a bracket at the moment, I'm not sure if that would change anything.