I understand that Kronecker Delta is a function that is defined for discrete arguments as follows:
$$\delta_{ij} = \begin{cases} 0, & i \ne j \\ 1, & i = j \end{cases}$$
However, I want to define a continuous function that as a limit behaves like Kronecker Delta. So I start with a Unit Triangle function $$T(x) = \begin{cases} 1-|x|, & |x| \lt 1 \\ 0, & \text{otherwise} \end{cases}$$
Then I define this continuous function as a limit like: $$G(x) = \lim\limits_{k \to \infty} (T(x))^k$$
I want to take continuous time Fourier Transform of $G(x)$. I am able to do that for specific small values of $k$, but the expression keeps growing and I am unable to simplify. Need help with hopefully finding a simple closed form of this FT, as k approaches infinity.