However, even under more general conditions versions of the Fourier inversion theorem hold.
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Conditions under which this inversion is valid are given in the Mellin inversion theorem.
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In applications of the Fourier transform the Fourier inversion theorem often plays a critical role.
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The most common statement of the Fourier inversion theorem is to state the inverse transform as an integral.
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For example, the Fourier inversion theorem on shows that the Fourier transform is a unitary operator on.
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The Mellin inversion theorem then shows that \ tilde { \ mathcal { M } } is invertible with inverse
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This version of the theorem is used in the proof of the Fourier inversion theorem for tempered distributions ( see below ).
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More abstractly, the Fourier inversion theorem is a statement about the Fourier transform as an Fourier transform on function spaces ).
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When used in physics and engineering, the Fourier inversion theorem is often used under the assumption that everything " behaves nicely ".
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The Fourier inversion theorem holds for all continuous functions that are absolutely integrable ( i . e . ) with absolutely integrable Fourier transform.
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