# Approximation to the $n$-th derivative using reproducing kernels.

For integrable functions defined on the real line, the normalized gaussian function approximates the convolution identity, Dirac Delta, in the sense that if $$g(t):=N_0e^{-x²}$$ (denoting the normalizing constant by $$N_0$$) and $$g_{a0}(x):=aN_0e^{-(ax)²}$$ then, for all $$f$$ integrable, $$\lim_{a\to\infty} f*g_{a0}=f$$ almost everywhere.

My first question is: Is this true?

My second question is: does the $$n$$-th derivative of the Gaussian function approximates the derivade operator by convolution? I mean writing $$g_{an}:=\frac{d^n}{dx^n} g_{a0}(x)$$ then $$\lim_{a\to\infty} f*g_{an}=f^{(n)}$$ and (third question) whether the $$n$$-th derivative operator includes or not the normalization constant $$N_n$$ for $$g_{an}$$.