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I am reading the notes Lecture notes on representation theory.

I have some difficulty in proving a) in Exercise 1.2. We need to prove that $$ \mathcal{F}^2(f)(x) = qf^{-}. $$

I compute as follows: $$ \mathcal{F}^2(f)(x) = \sum_{y\in k} \bar{\psi}(xy)\mathcal{F}(f)(y) \\ = \sum_{y\in k} \bar{\psi}(xy) \sum_{z\in k} \bar{\psi}(yz)f(z) \\ = \sum_{z \in k} \sum_{y \in k} \bar{\psi}(xy)\bar{\psi}(yz) f(z). $$

But why $\mathcal{F}^2(f)(x) = qf^{-}$? Thank you very much.

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