For a sequence of functions $g_n(x-x_o)$ approximating the Dirac delta I can write:
$ \int_a^b g_n(x-x_o) f(x) dx = \int_a^b \delta(x-x_o) f(x) dx + \epsilon_n$
when $x_o \in [a,b]$. I am trying to estimate the error term $\epsilon_n$ when $g_n(x-x_o)$ is a Gaussian function. Does anybody know how this might be done?
PS: I have seen calculations of upper bounds for $\epsilon_n$ when $g_n$ is a Gaussian. But these upper bound calculations are more concerned with the dependence of the error upon $n$ (which is inversely proportional to the width of the Gaussian) than a reasonable estimate of $\epsilon_n$. If I try using the error term given by these upper bounds calculations to estimate the error then the error is far larger than what one might expect.