On Peskin & Schroeder's QFT, page 30, the scalar field propagator as the retarded Green function is defined as
$$(\partial^2+m^2)D_R(x-y)=-i\delta^4(x-y) \tag{2.56}$$ The Fourier transformation is given as $$D_R(x-y)=\int \frac{d^4p}{(2\pi)^4}e^{-ip\cdot(x-y)}\tilde{D}_R(p) \tag{2.57}$$ with $$\tilde{D}_R(p)=\frac{i}{p^2-m^2} $$
My question is, does the inverse propagator Fourier transform is consistent in the closed form? $$D_R(x-y)^{-1}=\int \frac{d^4p}{(2\pi)^4}e^{-ip\cdot(x-y)}\tilde{D}_R(p)^{-1} \tag{*} $$
My answer to this question is not. However, I am asking this question since in later chapter 11, eq.(11.97) about the effective potential $$ \tilde{D}^{-1}\left(p^2\right)=\int d^4 x e^{i p \cdot(x-y)} \frac{\delta^2 \Gamma}{\delta \phi \delta \phi}(x, y)=0 \tag{11.97} $$ the book seems uses the relation of Eq.~$*$, since $$ \left(\frac{\delta^2 \Gamma}{\delta \phi_{\mathrm{cl}}(x) \delta \phi_{\mathrm{cl}}(y)}\right)=i D^{-1}(x, y) \tag{11.90} $$ where this relations arises from (11.89) and (11.87) $$ \begin{aligned} \delta(x-y)&=\int d^4 z \frac{\delta^2 E}{\delta J(y) \delta J(z)} \frac{\delta^2 \Gamma}{\delta \phi_{\mathrm{cl}}(z) \delta \phi_{\mathrm{cl}}(x)} \\ &=\int d^4 z D(y,z) D(z,x)^{-1}. \end{aligned} \tag{11.87} $$ and $$ \left(\frac{\delta^2 E}{\delta J(x) \delta J(y)}\right)=-i\langle\phi(x) \phi(y)\rangle_{\mathrm{conn}} \equiv-i D(x, y) \tag{11.89} $$
My thoughts:
(1) The relation of $F(p)=\frac{1}{2\pi}\int dx\ e^{-ipx} f(x)$ and $F^{-1}(p)=\frac{1}{2\pi}\int dx\ e^{-ipx} f^{-1}(x)$ should not coincide with each other for a general function of $f$;
(2) However, in our case of Green function, I am not sure. Can we infer the $D_R(x-y)^{-1}$ from (2.56)?
Edits:
Qmechanic's answer really make sense. But I still don't clear for
(3) Why $D^{-1}(x-y)=i(\partial^2+m^2)\delta^{(4)}(x-y) $? When I put this expression and Eq.(2.56) into Eq.(11.87), I don't obtain the correct form of $\delta(x-y)$.
(4) Why $D(x-y)$ is not invertible? Would this the same reason with the trivial photon propagator? (Like P & S Eq.(9.52)) \begin{equation} \begin{aligned} \left(\partial^2 g_{\mu \nu}-\partial_\mu \partial_\nu\right) D_F^{\nu \rho}(x-y) & =i \delta_\mu{ }^\rho \delta^{(4)}(x-y) \\ \text { or } \quad\left(-k^2 g_{\mu \nu}+k_\mu k_\nu\right) \widetilde{D}_F^{\nu \rho}(k) & =i \delta_\mu{ }^\rho, \end{aligned} \tag{9.52} \end{equation} where the Feynman propagator $D^{\nu \rho}_F$ has no solution, since the $4\times 4$ matrix $(-k^2g_{\mu \nu}+k_{\mu} k_{\nu})$ is singular.
