The Fourier transform existed for the sake of existing for ~200 years before it turned out to be useful for building the entirety of our communications infrastructure on top of.
In cold hard dates, though, Fourier was already using his series to solve heat transfer problems in 1822.
I don't agree with the bizarre idea that every bit of research should have a clear utility. I'm just being careful about dates. And I think FTs kind of were invented with a view towards solving differential equations for physics. Just not electrical ones.