Relation between the Fourier transform of Y and X
I too have struggled with this notation. It seems to defy logic that a j would appear in the brackets, it almost appears that j is an argument as opposed to just a number. Regardless, this notation is seen frequently in the literature so try to get used to it. It almost seems like they are taking the Laplace transform and evaluating it at s=jw which would make sense. I don't know exactly the reason for the different notations. Traditionally, as a function, the only argument is w and not jw so in that sense the j shouldn't be there. But just try to get used to it as it is written like this a lot.