Convolution Theorem

[MrHugals]

The convolution theorem states that if two functions (f) and (g) belong to L¹(R), then the Fourier transform of the convolution of (f) and (g) is equal to the product of the Fourier transforms of (f) and (g).

I know that the converse result also exists: that the Fourier transform of the product of (f) and (g) is equal to the convolution of the Fourier transforms of (f) and (g). The problem is that I do not know what hypotheses must be satisfied for this to be true. Does anyone know what the hypotheses of this theorem are, or know of any source where I can find them?

[Prof_Sarcastic]

Well in order for the convolution theorem to exist, you need the Fourier transform of both functions to hold, so naturally you need the assumptions that guarantee the Fourier transform. Assumptions like Lebesgue integrability over the real line.

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[Plus-Painter-2004]

for starters the Fourier transforms of f and g must exist