Yes, all true, and well known to me. What seemed so intriguing, due to a misunderstanding, was the ide that you had explained to someone "Here's what the Fourier transform is about, and here's how you do it, and let me explain it to you by starting with convolutions." That would be something I'd like to know more about: start with convolutions and use them to explain Fourier and/or Euler. But that's not what you meant.
Incidentally, in the BBCode used here, the key to displaying superscripts and subscripts is to NOT use
Code:
tags. [Code] tags disable all the other tags that you use to format the text. So, with [code] tags I'll get
[code]∫f(t)∙e[sup]-𝒋𝛚t[/sup]dt
and when I remove those I get
∫f(t)∙e
-𝒋𝛚tdt