Fourier transform is so fundamental to understanding the world and mastering Fourier transform (and linear algebra too) is so sooo important before entering anything "quantum". Quantum field theory (QFT), especially, is essentially a way to look at the world through its Fourier transform.
In a nutshell Fourier transform, the derivative operator, exponential function and CAUSALITY are related at a very deep level. And that's why it's such an important mathematical tool. Because it is related to causality.
I understand all the noted mathematics. Any suggestions for references on learning quantum theories? For most materials, I get stuck as they use notations unfamiliar to me in spite of me being familiar with the underlying mathematics.
You could greatly benefit from Prof. Leonard Susskind's lectures on quantum mechanics and QFT on Youtube (Standford university's channel), which are very accessible. A great physicist who happens to be a great teacher and who gets all the deep interconnections involved. This is one playlist (he has many more): https://www.youtube.com/watch?v=JzhlfbWBuQ8&list=PL84C10A9CB...
In a nutshell Fourier transform, the derivative operator, exponential function and CAUSALITY are related at a very deep level. And that's why it's such an important mathematical tool. Because it is related to causality.