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Mathematical Physics Question on Fourier Transform

The Fourier transform and its inverse transform are respectively defined as:f~(ω)=12π∫−∞+∞f(x)eiωxdx\tilde{f}(\omega) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} f(x) e^{i \omega x} dxandf(x)=12π∫−∞+∞f~(ω)e−iωxdωf(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} \tilde{f}(\omega) e^{-i \omega x} d\omegaConsider two functions ff and gg. Another function f∗gf * g is defined as:(f∗g)(x)=12π∫−∞+∞f(y)g(x−y)dy(f * g)(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} f(y) g(x - y) dyWhich of the following relation is/are true?Note: Tilde (~) denotes the Fourier transform.

A

f∗g=g∗ff * g = g * f

B

f~∗g=g∗f~\tilde{f} * g = g * \tilde{f}

C

f~∗g=f~g\tilde{f} * g = \tilde{f} g

D

f~∗g=f~g~\tilde{f} * g = \tilde{f} \tilde{g}

Answer

f∗g=g∗ff * g = g * f

Explanation

Solution

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