Quantum mechanics and QFT use extensively Fourier analysis.
When trying to approximate a periodic function by Fourier series (say a rectangular wave), it is possible to increase the number of terms until the approximation seems good enough for our ends.
But Fourier analysis aims to approximate also functions limited to a finite range. The underlying idea is to use the destructive interference of the harmonic functions to get zero out of that range.
Suppose I know $f(p)$, and want to evaluate numerically, (because the integrand is not analytically solvable), the Fourier transform to get $f(x)$.
I can proceed as in Fourier series, choosing an interval, splitting $f(p)$ in a series of coefficients that multiplies the exponentials, and adding all the parts.
But when I go that way, what I get is always a periodic function, what I know it is not the case.
Is there other way to have progressively better estimates as it happens in Fourier series (or Taylor expansions)?