For fixed \(x^{\prime}\), the Gaussian kernel function is the solution to Fourier's heat equation \[ \frac{\partial}{\partial t} f(x \mid t)=\frac{1}{2} \frac{\partial^{2}}{\partial x^{2}} f(x \mid t), \quad x \in \mathbb{R}, t>0 \]
with initial condition \(f(x \mid 0)=\delta\left(x-x^{\prime}\right)\) (the Dirac function at \(x^{\prime}\) ). Show this. As a consequence, the Gaussian KDE is the solution to the same heat equation, but now with initial condition \(f(x \mid 0)=n^{-1} \sum_{i=1}^{n}\left(x-x_{i}\right)\). This was the motivation for the theta KDE [14], which is a solution to the same heat equation but now on a bounded interval.