# Is an interval for a function and its Fourier transform based on the time constants?

The Fourier transform of an exponential function is a Lorentzian. For the sum of multiple exponential functions with time constants $$k$$, is it only meaningful to define the function and its Fourier transform on an interval given by the minimum and maximum time constant, since these values set the time scales for the system? Although measurements can be made outside this interval, I don't know if it's meaningful to look at theoretically (i.e., to extrapolate this function in those cases). Hoping someone can shed some light onto this.

Certainly not. I think that would be a major mistake. The function is well defined for all $$t$$ and has lots of energy outside the interval $$[k_{min},k_{max}]$$. In fact it's maxima are at $$t=0$$ and $$f = 0$$. Restricting any analysis $$[k_{min},k_{max}]$$ will give substantially different results than analyzing the complete function. By that definition I would say it's definably meaningful (and important) to look at the whole thing.