Two texts I keep on my desk that are a bit more rigorous than most:

- Oppenheim & Schafer Discrete Time Signal Processing (the Bible of DSP)

- Manolakis & Ingle, Applied Digital Signal Processing (good discussion of orthogonal transforms).

A lot of what I know about transform analysis comes from self study of linear algebra/vector spaces with some reading here and there in commonly cited papers. Might want to pick up a text on that subject, it's the same idea but more rigor than an engineer would use.

There's also a book I haven't worked all the way through yet and is dated, heavily based in EE concepts (non negligible amount of circuit theory) and extremely rigorous called Theory of Linear Physical Systems by Ernst Guilleman. I picked it up last week actually and quite like it.

It has a lot of information and approaches with Fourier/Laplace methods, which is interesting since it predates the FFT and has so much information on concepts that engineers 50 years ago would need to build their intuition with instead of through tooling. I picked it up for the network theory/dynamical systems angle (which relates to some stuff I'm working on) but the rigor is definitely higher than what you'd see in those more digestible books.

Practical Signal Processing by Owen might be a lead worth looking into. It was recommended by Ossmann in his HackRF tutorial videos (he created the HackRF).

It's gonna fall near the extreme end of succinctness.

I had since purchased the book but haven't gone more than a couple chapters deep. It definitely looks like it allows you to dive into the meat of DSP. Don't know if it will fit the bill for rigorous.

I also didn't pay anything near what it's currently listing for on Amazon.

You could try Arfken and Weber's "Mathematical Methods for Physicists". It's a common reference for Physicists. While it doesn't focus on engineering (or signal) applications, it's both fairly rigorous and succinct enough for practical use. Looks like there are pdf's online.