Since you mentioned that you are a math student, I highly recommend Stein and Shakarchi's "Fourier Analysis: An Introduction".

It is a rigorous treatment that covers a lot of the essentials of Fourier theory, while also covers some very interesting "math applications" (e.g Dirichlet's theorem on primes in arithmetic progressions at the end), as well as "mixed applications" (e.g Radon transforms that are used in imaging but are also of interest from a math perspective).

Note that this book requires an introductory analysis class as a prereq; basically you need to be familiar with the standard Riemann integral theory and rigorous treatment of limits, continuity, and derivatives.