I was reading this article in wikipedia, and I supposed $f,g \in L^1(\mathbb{R^n})$ such that their product $f \cdot g$ are in $L^1(\mathbb{R^n})$ too. So let $h=f \cdot g$, and $\hat{h}(x)=\int_{\mathbb{R^n}} [f(y)\cdot g(y)]e^{-ixy}dy$. On other hand, $\hat{f}(y)=\int_{\mathbb{R^n}} f(z)e^{-iyz}dz$ and $\hat{g}(x)=\int_{\mathbb{R^n}} g(y)e^{-ixy}dy$, And we know for inverse Fourier transform: $$f(y)=\frac{1}{2\pi}\int_{\mathbb{R^n}} \hat{f}(w)e^{iwy}dw$$
Even more $$\hat{h}(x)=\int_{\mathbb{R^n}} [(\frac{1}{2\pi}\int_{\mathbb{R^n}} \hat{f} (w)e^{iwy}dw) g(y)]e^{-ixy}dy$$ And using Fubini: $$\hat{h}(x)=\frac{1}{2\pi}\int_{\mathbb{R^n}}\hat{f}(w)\hat{g}(x-w)dw =\frac{1}{2\pi}(\hat{f}*\hat{g})(x)$$
And I don't understand. Why is different to the result of wikipedia? my result is incorrect?