I can't help but think of orthogonal frequency-division multiplexing and it's use in encoding data on multiple carrier frequencies, and it makes me wonder what other parallels we will discover between digital transmission technology for cross-domain stuff like this.

Not even cross-domain. (Nor cross-co-domain.)

Trigonometric polynomials are also polynomials. And linear spaces are all "the same". That is what the definition is for. Even the transpose-mapping is linear.