Hilbert, Bochner, and Studying for Amateur Extra

I’ve been studying for the Amateur Extra license exam, the highest level of FCC license for amateur radio. Right now, I’m reading about Hilbert transforms because they are mentioned in a question (E7C09 from the Extra question pool, if you care). I’m pretty sure that I learned Hilbert transforms thirty years ago in my Signal Theory class.

Oddly, my sole memory from that class is about Salomon Bochner. Bochner had come to Rice to take a chair in mathemetics after retiring from Princeton. Checking his bio, he’d retired just after I arrived at Rice, though he ran the department for several years, so that alone gets some respect.

During one lecture in the signal theory class, we hit a total wall. We needed something to be tractable with one of the hammers that we had in our toolbox, but we couldn’t tell if it was. The professor whipped out Bochner’s Theorem, where if the function was positive definite, then our best tools were available. It was like magic. Math, it works when the hammer doesn’t.