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I have seen that if Fourier operator is defined by

$$ h(k) = \hat F(g(x)) = \frac{1}{\sqrt{2\pi}}\int_{- \infty}^{\infty} dx\:g(x)\:e^{ikx} $$ then $$ \hat F^2\{g(x)\}=g(-x) \implies \hat F^2 \equiv \hat P $$ where $ \hat P $ is the parity operator.

If this is the case I can also show that $ \hat F^4 \equiv \hat I $ where $ \hat I $ is the identity operator. But I am not able to prove the first relation relating fourier and Parity operator. Also would this follow that higher roots of fourier operator can be reduced any one of these three operators (that is $ \hat P $, $ \hat I $ and $\hat F$)?

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