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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 $T(R_k z)=R_k T(z)$

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    (i) to get the composition operator, use `\circ`, not `o`. (ii) You shold tell us what you've tried. You have posted 22 questions, all of them fairly clearly homework assignments. A quick random sampling shows you seldom, if ever, give anything except the statement. Moreover, you have only accepted 4 answers so far even though 18 questions have been answered. This site is not meant to be simply a "help-me-do-my-homework" site, and you are treating people's time rather cavalierly.2010-11-22

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Some advices:

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.

When you attend lectures, try to study the chapter before the lecture.