= Solution
The <convolution theorem> turns the product $e^{-k^2t}\widetilde\theta(k,0)$ into a spatial <convolution>. Its kernel is
$$
G(z,t)=\frac1{2\pi}\int_{\mathbb R}e^{-tk^2+ikz}dk,\qquad t>0.
$$
Complete the square:
$$
-tk^2+ikz=-t\left(k-\frac{iz}{2t}\right)^2-\frac{z^2}{4t}.
$$
After rescaling the variable, the supplied translated <Gaussian integral> gives
$$
\boxed{G(z,t)=\frac1{\sqrt{4\pi t}}e^{-z^2/(4t)},\qquad
\theta(x,t)=\frac1{\sqrt{4\pi t}}\int_{\mathbb R}\theta(y,0)e^{-(x-y)^2/(4t)}dy.}
$$
This is the <heat-kernel solution> for diffusivity one. Its kernel integrates to one, preserving the total temperature integral, and approaches a <Dirac delta function> as $t\downarrow0$. Its variance is $2t$; the factor is appropriate to $\theta_t=\theta_{xx}$, rather than the alternative convention with diffusivity $1/2$.
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