If $u$ is a solution of the equation $$\frac{\partial}{\partial t} u(x,t) + \int_{-\infty}^{\infty} \text{sinc}(x-y) \cdot \frac{\partial^{2}}{\partial y^{2}} u(y,t) \ dy = 0,$$ with initial condition $\ u(x,0) = f(x).$ Let $U^{t}(x) = u(t,x)$. How can I find an expression for the Fourier transform ${\widehat{U}}^{t}(k)$?
I'm not sure how to take Fourier transform inside the integral.
The integral term of the equation is the convolution of the $\operatorname{sinc}$ function and the second derivative of $U^t(x)$. Taking the Fourier transform of the equation and using its properties we get $$ \frac{d}{dt}\hat U^t(k)+\widehat{\operatorname{sinc}}(k)\bigl(-C\,k^2\,\hat U^t(k)\bigr)=0, $$ where the constant $C$ depends on the definition of Fourier transform you are using. For each $k$ fixed this is a first order linear differential equation that you can solve to obtain $\hat U^t(k)$.