Problem $38$: $(18$ points$)$

In the first problem you will prove a series of Fourier transform properties that will be used extensively in the remainder of this problem set. Given that

$f(t)>F()$

$g(t)>G()$

and $t_o$ and ${}_o$ are real$-$valued constants, derive the following Fourier Transform properties:

$(3$ points$)$ Symmetry Property

$F(t)>2f(-)$

$(2$ points$)$ Time Shift Property

$f(t-t_o)>F()e^{-t_0}$

$(2$ points$)$ Frequency Shift Property

$f(t)e^{{}_{o}t}>F(-{}_o)$

$(4$ points$)$ Time Convolution

$f(t)**g(t)>F()G()$

$(4$ points$)$ Frequency Convolution

$f(t)g(t)>\frac{1}{2}F()**G()$

$(3$ points$)$ Time Differentiation

$\frac{d^{n}f}{d{t}^n}>({)}^{n}F()$