Before giving a more detailed question below, the basic one is: can anyone recommend a good signal-processing reference which would be maximally readable by a pure mathematician (who nevertheless wants to use the techniques for actual data analysis)?

Actually, the specific thing I want to understand is how to analyze a small segment of a mystery signal, which should just be the sum of a few sinusoids, but for which the available segment is potentially much shorter than the period of the sinusoids. The goal is to recover the underlying frequencies. I'd appreciate a general reference anyway, but if someone can address my specific problem, that would help too, of course!

More detail: I am a pure mathematician, but I need to learn some signal processing techniques for a side-project in biology. At first I was excited by this, since I am already comfortable with all the "underlying" background material (essentially just Fourier transforms, etc.) and thought it would be easy enough and fun to grasp what engineers and scientists were actually doing.

However, all references I've found are written for someone with the opposite background, or at least, they are written not just to deemphasize math, but sort of to avoid it as much as possible. This makes it very hard and a bit frustrating for me to read, since firstly a bunch of terminology is thrown at me without an underlying theory (so many specific window functions with different names!) and the fact that I am very comfortable with analysis hasn't helped much. My first instinct was just to go at it alone from first principles (the definition of the DFT...) but it seems silly to ignore the vast history in this subject.

I did not mean that to sound as ranty as it did. I am not complaining, I am just wondering if anyone can recommend a reference more amenable to my background, which could take advantage of it or at least, ease me into the subject.

Thanks!