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In page Fourier transform:

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The generalization of the Fourier transform to the noncommutative situation has also in part contributed to the development of noncommutative geometry.[citation needed] In this context, a categorical generalization of the Fourier transform to noncommutative groups is Tannaka–Krein duality, which replaces the group of characters with the category of representations. However, this loses the connection with harmonic functions.