1. **Stating the problem:** We want to establish the relationship between the convolution of the Fourier transforms $F(f) * F(g)$ and the Fourier transform of the product $F(fg)$, given that $f$ and $g$ are integrable functions. 2. **Recall the definitions and properties:** - The Fourier transform of a function $h$ is defined as $$F(h)(\\omega) = \int_{-\\infty}^{\\infty} h(t) e^{-i \\omega t} dt.$$ - The convolution of two functions $a$ and $b$ is defined as $$ (a * b)(\\omega) = \int_{-\\infty}^{\\infty} a(\\xi) b(\\omega - \\xi) d\\xi.$$ 3. **Key property (Convolution Theorem):** The Fourier transform of the product of two functions $f$ and $g$ is the convolution of their Fourier transforms: $$F(fg)(\\omega) = (F(f) * F(g))(\\omega) = \int_{-\\infty}^{\\infty} F(f)(\\xi) F(g)(\\omega - \\xi) d\\xi.$$ 4. **Explanation:** - Multiplying two functions in the time domain corresponds to convolving their Fourier transforms in the frequency domain. - This is a fundamental property used in signal processing and analysis. 5. **Summary:** The convolution of the Fourier transforms $F(f) * F(g)$ equals the Fourier transform of the product $fg$, i.e., $$F(fg) = F(f) * F(g).$$ This establishes the connection between the two operations.