At your level, you might think of the eigenfunctions of your time-independent hermitean hamiltonian analogously to the exponential kernels of the Fourier transform. That is, if your eigenfunctions are [complete](https://physics.stackexchange.com/questions/251803/what-is-meant-by-the-term-completeness-relation), $$ \sum_n \psi(x) \psi^*(y) = \delta(x-y)~~~~\leftrightarrow ~~~~\sum_n |\psi_n\rangle \langle \psi_n|= \mathbb{I}, $$ you may transform from one space to the other, assuming a swamp of fussbudget qualifications, of which this site is full. That is, $$ |\psi_n\rangle = \int\!\!dx ~ ~\psi(x) |x\rangle ~~~~\leftrightarrow ~~~~|x\rangle= \sum_n \psi_n^*(x)|\psi_n\rangle , $$ the latter of which might have an elegant summation or not, as in the textbook case of the [SHO](https://physics.stackexchange.com/questions/405060/how-does-one-calculate-the-position-eigenvalues-of-the-matrix-corresponding-to-t). In any case, the complete orthonormal set of eigenfunctions of the hamiltonian are some type of transform you might analogize to the Fourier one...