91Ó°ÊÓ

In Exercises 21-26, use Exercise 20 and known transforms to compute the Fourier transform of the given function. $$ f(x)= \begin{cases}\cos x & \text { if }|x|<1_{1} \\ 0 & \text { otherwise }\end{cases} $$

In Exercises \(29-46\), find the Fourier transform of the given function. To simplify your computations, use known transforms and operational properties. $$ \phi(x)=-\delta_{-1}(x)+\delta_{1}(x)+U_{0}(x)-U_{1}(x) $$

In Exercises \(29-46\), find the Fourier transform of the given function. To simplify your computations, use known transforms and operational properties. $$ \phi(x)=x\left(U_{-1}(x)-U_{1}(x)\right). $$

In Exercises \(37-46\), use the operational properties and a known Fourier transform to compute the Fourier transform of the given function: $$ f(x)=x e^{-x^{2}} $$

Use the operational properties and a known Fourier transform to compute the Fourier transform of the given function: $$ f(x)=x^{2} e^{-|x|} $$

Use the operational properties and a known Fourier transform to compute the Fourier transform of the given function: $$ f(x)= \begin{cases}x & \text { if }|x|<1 \\ 0 & \text { otherwise }\end{cases} $$

Use the operational properties and a known Fourier transform to compute the Fourier transform of the given function: $$ f(x)=\frac{x^{2}}{\left(1+x^{2}\right)^{2}} $$

In Exercises \(29-46\), find the Fourier transform of the given function. To simplify your computations, use known transforms and operational properties. \(f(x)=e^{-x}\) if \(x>0\) and 0 if \(x<0\)

Use the operational properties and a known Fourier transform to compute the Fourier transform of the given function: $$ f(x)=\left(1-x^{2}\right) e^{-x^{2}} $$

In Exercises \(29-46\), find the Fourier transform of the given function. To simplify your computations, use known transforms and operational properties. \(f(x)=e^{-2}\) if \(|x|<1\) and 0 otherwise.