Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 324 / 1 / b / Solution

A shift-invariant basis is a common eigenbasis of all cyclic shift operators $U(\gamma )$. For the displayed Fourier states, orthonormality follows from the root-of-unity filter:There are $N$ vectors in this orthonormal set in the $N$-dimensional Hilbert space, so they form a basis. Reindexing the finite sum givesThus every $|{\xi }_{\alpha }⟩$ is simultaneously an eigenvector of every shift, with eigenvalue ${\omega }^{\alpha \gamma }$ for $U(\gamma )$.