At the end of some calculations I've reached
$$\lim \limits _{t \to 0_+} \int \limits _{\Bbb R ^n} \frac {h(t,x,y)} t f(y) \Bbb d y$$
where
$$h(t,x,y) = \frac {\Bbb e ^{\frac {\Bbb i |x-y|^2} {4t}}} {\sqrt {4 \pi t}^n}$$
is the imaginary heat kernel. Now, I know that without that $t$ in the denominator, the limit would be $f(x)$. With it in the denominator, can one give a rigorous meaning the the above formula? More generally, can one give a meaning to
$$\lim \limits _{t \to 0_+} \int \limits _{\Bbb R ^n} \frac {h(t,x,y)} {t^p} f(y) \Bbb d y$$
when $p \in \Bbb N \setminus \{0\}$?
Furthermore, can $h$ be written as a convergent power series in $t$ in $\mathcal D ' (\Bbb R ^n)$ with the first term $\delta_x$ (even though I have never encountered the concept of power series in a locally-convex space)? If such an expansion is true only asymptotically, can the coefficients be explicitly written?
If the original question is too complicated, I would be satisfied even with an answer for $h(t,x,y) = \frac {\Bbb e ^{- \frac {|x-y|^2} {4t}}} {\sqrt {4 \pi t}^n}$ (note the $-1$ instead of $\Bbb i$). For convenience, feel free to take $n=1$.