Suppose $T_1, T_2 : \ell^2(Z_N)-> \ell^2(Z_N)$ are translation invariant linear transformation. How to prove that the composition $T_2 \circ T_1$ is translation invariant? Recall the translation operator $R_k$ defined by $(R_k z)(n)=z(n-k) $ Also recall definition: Let $T : \ell^2(Z_N)-> \ell^2(Z_N)$ be a linear transformation. $T$ is translation invariant if $T(R_k z)=R_k T(z)$
In general, try to break down the problem into pieces you are familiar with and understand - if you feel uncomfortable with some part, then you should study that again and again.
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