A path to get the foundation of SP and DSP for a pure math
You have already read those Oppenheim's Signals &amp; Systems, and Discrete-Time Signal Processing books.
I'm not sure what you mean by foundations but in some sense these two are also the foundations on signal processing. In other words, there are no (popular &amp; successful) graduate level DSP books that discuss at an advanced level the same topics that are covered on them.
However the following books (or subjects) will enhance your understanding, or broaden your appreciation of the subject.
First of all, the very first graduate level course on DSP, Communications, and Control is called Linear System Theory which brings together all the undergraduate mathematical stuff from a new, advanced, deeper, and foundational point of view of the Hilbert (linear vector) Spaces, Linear Mappings, and Matrix theory. It does not have a definite book but a bunch of books on Linear Algebra &amp; Matrices, Measure Theory, and Diferential Equations were used. Note that there's a control theory oriented bunch of Linear System Theory books (Desoer's crew) that I do not recommend for DSP, unless you will be designing control systems on the field. Signal processing does not make much use of state-space approach, unless it's absolutely necessary.
Then the second refresher/deepener is on Probability, Statistics and Random Processes. Fortunately it has two very strongly recomended books though:

Statistical Digital Signal Procesing _ Monson HAYES
Discrete RaNdom Signals and Statistical Signal Processing _ THERRIEN

The first book is a must read, the second one is following the style of Oppenheim series but is harder to follow, and less practical than the first.
Then the following books / subjects will be awating you :

Adaptive Filter Theory _ HAYKIN
Multiresolution Signal Decomposition _ AKANSU
Multirate Digital Signal Processing _ RABINER
Estimation &amp; Detection Theory _ KAY
Pattern Classification _ DUDA
Theory and Applications of Digital Signal Processing _ RABINER &amp; GOLD

Then the following applications will make your day :

Speech and Hearing for Communication - FLETCHER

Digital Processing of Speech _ Rabiner

Speech and Audio Signal Processing _ GOLD

Discrete-Time Processing of Speech _ PROAKIS

Advances in Speech Coding _ GERSHO

Two-Dimensional Signal and Image Processing _ LIM

Digital Image Processing _ GONZALES

Fundamentals of Image Processing _ JAIN

Signal Compression_JAYANT

Introduction to Data compression _ SAYOOD


Of course the list is by no means complete...