Let $f\in \mathcal{L}(\mathbb{R}^1).$ Prove that $$f(x)=\frac{1}{\sqrt{2\pi}}\lim_{T\to+\infty}\frac{1}{T}\int_{0}^{T}\int_{-t}^{t}e^{ixy}\hat fdy\,dt$$ for almost all x including points of continuity of f.
I tried to repeat some steps from other inversion theorem proofs. $$\int_{0}^{T}\int_{-t}^{t}e^{ixy}\hat f(y)dy\,dt = \int_{0}^{T}\int_{-t}^{t}e^{ixy}\int_{-\infty}^{\infty}e^{-izx}f(z)dz\,dy\,dt = \int_{0}^{T}\int_{-\infty}^{\infty}f(z)\int_{-t}^{t}e^{iy(x-z)}dy\,dz\,dt = 2\int_{0}^{T}\int_{-\infty}^{\infty}f(z)\frac{\sin{t(x-z)}}{(x-z)}dz\,dt$$ That's where I stuck.
I would be happy to get some hints.