I am given the following heat equation: \begin{equation} u_t(x,t)-\Delta u(x,t)=0 ~~~ x \in \mathbb{R}^n,~t>0 \end{equation} with the initial condition $u(x,0)=g(x)=\sum_{j=1}^m j \delta(x-x_j)$ which I would like to solve by using the Fourier transform.
At first, we can transform both sides. This leads to the following ODE: \begin{equation} \hat{u}(\xi,t)=\hat{g}(\xi)e^{-|\xi|^2t} \end{equation}
Now using the convolution theorem and taking the inverse on both sides we derive \begin{equation} u(x,t)=\mathscr{F}^{-1} (\hat{g}(\xi)e^{-|\xi|^2t}) = (2\pi)^{n/2} g(\xi) \ast \mathscr{F}^{-1}(e^{-|\xi|^2t}) = g \ast K \end{equation}
where $K(x,t)=(4 \pi t)^{n/2} e^{-|x|^2/4t}$.
In other words, we have that \begin{equation} u(x,t)=\frac{1}{\sqrt{4 \pi t}^n} \int_{\mathbb{R}^n} g(y) e^{\frac{-|x-y|^2}{4t}}dy. \end{equation}
What I am stuck now with is how to compute an explicit solution for my initial condition $g(x)=\sum_{j=1}^m j \delta(x-x_j)$ as I do not know how to compute the integral
\begin{equation} \int_{\mathbb{R}^n} \delta(y-y_j)e^{\frac{-|x-y|^2}{4t}}dy. \end{equation}
I am very new to the topic of dirac-deltas so I am not sure what to do with them in an integral.
My second approach was to first compute the Fourier transform of $g$, namely
\begin{equation} \mathscr{F}(\delta(x-x_j)) = \frac{1}{\sqrt{2\pi}^n} \int_{\mathbb{R}^n} \delta (x-x_j) e^{iyx} dx \stackrel{?}{=} \frac{1}{\sqrt{2\pi}^n} e^{iyx_j}? \end{equation}
Could someone help me how to proceed further
