\displaystyle \begin{aligned} (\hat{g}\ast\hat{f})(\xi)&=\dfrac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}\hat{g}(y)\hat{f}(\xi-y)\mathrm{d}y \\ &=\dfrac{1}{\sqrt{2\pi}}\int_{-2}^{2}\dfrac{\sqrt{\pi}}{2\sqrt{2}}(2-|y|)\hat{f}(\xi-y)\mathrm{d}y...