I have no idea about engineering, but I don't see how a physicist would not know this. Basis transformations in infinitely dimensional Hilbert spaces are an int

I have no idea about engineering, but I don't see how a physicist would not know this. Basis transformations in infinitely dimensional Hilbert spaces are an integral part of quantum mechanics. The Fourier-transform in particular is used for transformation between space |r> and momentum |p> representations, but it's also common to use energy representations |E_n> where the orthonormal basis is typically somewhat more exotic (e.g. given by a Gaussian multiplied with Hermite polynomials, as is the case for the QHO).