Question

The Fourier transform and its inverse transform are respectively defined as:\[ \tilde{f}(\omega) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} f(x) e^{i \omega x} dx \]and\[ f(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} \tilde{f}(\omega) e^{-i \omega x} d\omega \]Consider two functions \( f \) and \( g \). Another function \( f * g \) is defined as:\[ (f * g)(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} f(y) g(x - y) dy \]Which of the following relation is/are true?Note: Tilde (~) denotes the Fourier transform.

• A. \( f * g = g * f \)
• B. \( \tilde{f} * g = g * \tilde{f} \)
• C. \( \tilde{f} * g = \tilde{f} g \)
• D. \( \tilde{f} * g = \tilde{f} \tilde{g} \)

Answer 1

The Correct Option is A, B, D

The correct Answers are(A): \( f * g = g * f \),(B): \( \tilde{f} * g = g * \tilde{f} \),(D):\( \tilde{f} * g = \tilde{f} \tilde{g} \)