A Problem on Tunneling in a dissipative environment

I am currently self-studying Many-Body Physics and I am using the textbook Condensed Matter Field Theory . I am currently trying to figure out the problem on page 151 (Chapter 3 and problem called "Tunneling in Dissipative Environment"). The action of the string is given by

$$S_{string} = int_{-infty}^{infty} dtau int_{-infty}^{infty} dx bigg[frac{rho}{2}dot{u}^2 + frac{sigma}{2}bigg(frac{partial u}{partial x}bigg)^2 bigg] leftrightarrow int_{-infty}^{infty} frac{domega}{2pi} int_{-infty}^{infty}frac{dk}{2pi} frac{rho omega^2 + sigma k^2}{2}|u(omega,k)|^2.$$

This part I understand; in order to jump to Fourier space, time is "promoted" to frequency and position is "promoted" to momentum $k$. Time derivatives would be replaced with $iomega$ and space derivatives are replaced with $ik$. The square of these would be $-omega^2$ and $-k^2$(perhaps the the above equation is missing a negative sign; or perhaps time derivatives $times$ time derivative could be $iomega$ $times$ ($iomega$)$^{*}$).

The textbook also claims the following:

$$delta(q(tau) – u(tau,0))= int Df e^{iint_{-infty}^{infty}dtau f(tau)big(q(tau) – u(tau,0)big)} leftrightarrow int Df expbigg[iint_{-infty}^{infty}frac{domega}{2pi} f(omega)(q(-omega) – int_{-infty}^{infty}frac{dk}{2pi}u(-omega, -k)) bigg].$$

Where did the expression $q(-omega) – int_{-infty}^{infty}frac{dk}{2pi}u(-omega, -k)$ come from? Where did the momentum space integral come from, since I do not see any space derivatives in the integral to the left? Also, why is it $q(-omega)$ instead of $q(omega)$?

Next, the text claims that

$$int Du e^{-S_{string} – iint dtau f(tau)u(tau,0)} propto expbigg[-int_{-infty}^{infty}frac{domega}{2pi}int_{-infty}^{infty}frac{dk}{2pi}frac{1}{2}frac{|f(omega)|^2}{rho omega^2 + sigma k^2} bigg].$$

How does does derive this?