I am curious if there is any definition or understanding of Fourier transform averaged with respect to time length to allow the transformation of non-integrable functions?
By time averaged I mean:
$F_{Tavg}({\omega}) = lim_{T\rightarrow \infty}\frac{F_{T}(\omega)}{T} = lim_{T\rightarrow \infty}\frac{1}{T}\int_{\frac{-T}{2}}^{\frac{T}{2}}e^{t\omega j}f(t)\chi_{[\frac{-T}{2},\frac{T}{2}]}dt$
This concept is somewhat used in the definition of Power Spectral Density but I am only interested in deterministic signals.