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Let $T: L^2(\mathbb{R}^n) \rightarrow L^2(\mathbb{R}^n)$ be the Fourier transform. Is there any reasonable definition of fractional Fourier transform (i.e. operator $A$ such that $A^{\alpha}=T$ for $\alpha \in (0,1)$), and if there is, is it of any use?

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closed as no longer relevant by Yemon Choi, S. Carnahan, Steve Huntsman, Andy Putman, Harald Hanche-Olsen Apr 9 '10 at 18:31

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Voting to close as "just ask Google or Wikipedia" (and lack of background motivation other than natural curiosity) – Yemon Choi Apr 9 '10 at 17:19