Fourier transform on Green's function

I need to do a Fourier transform for the next Green's function:$$F[(\frac{\mathrm{d} ^{2}}{\mathrm{d} x^{2}}-m^{2})\cdot G(x,x^{'})]$$.

My solution is: $$\int_{-\infty }^{\infty }(\frac{\mathrm{d} ^{2}}{\mathrm{d} x^{2}}\cdot G(x,x')\cdot e^{-ikx})dx-m^{2}\int_{-\infty }^{\infty }G(x,x')\cdot e^{-ikx}dx=\int_{-\infty }^{\infty }(\frac{\mathrm{d} ^{2}}{\mathrm{d} x^{2}}\cdot G(x,x')\cdot e^{-ikx})dx-m^{2}\widetilde{G}(k)$$

but I don't know how to solve the first integral.

The solution is:$$-(k^{2}+m^{2})\cdot \widetilde{G}(k)$$

Can someone please explain to me how did they get this expression?

• Looks related to math.stackexchange.com/questions/430858/… – Peter Jan 10 '20 at 9:06
• Use partial integration twice and then that G(x,x') should vanish for x to $\pm \infty$ – Peter Jan 10 '20 at 9:09
• Thank you all for the help, I have completely forgotten about differentiation property of the Fourier transform. – violettagold Jan 10 '20 at 9:12