Limit of The Dirac Comb

Question

The Dirac comb function with period T is: $$ f(t,T):=\sum_{k=-\infty}^{k=\infty}\delta(t-kT) $$

What is the limit of: $$ \lim_{T\to0} f(t,T) $$ ?

Answer 1

Intuitively it is clear that a very dense set of Dirac functions with constant amplitude, and properly normalized, behaves as the unit constant function. Therefore we expect that the limit of the (normalized) Dirac comb function for $T$ to zero is $f(t) = 1$.

Let us see if we can derive this result a bit more rigorously. Consider a Dirac comb function in the time domain with period $T$. Performing the Fourier transformation yields a Dirac comb function in the frequency domain with period $1/T$. We now take the limit of $T$ to zero. Only the Dirac function at $f=0$ remains, since all the others shift to plus or minus infinity, where they can be assumed to contribute no longer. Now perform the Fourier transformation back to the time regime on the remaining Dirac function. You get the constant function: $f(t) = 1$.