We discuss the convolution of two function over the real line and see that the Fourier Transform of a convolution is the product of the Fourier Transforms. This is a very powerful tool in Fourier Analysis. One consequence of this is the Plancherel Theorem. This states that the L^2 norm of the function is equal to the L^2 norm of the Fourier transform; namely, the Fourier transform is an L^2 isometry.
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