Σειρές Φουριέ: Διαφορά μεταξύ των αναθεωρήσεων

\left[f*_{2\pi}g\right](x) \ &\stackrel{\mathrm{def}}{=} \int_{-\pi}^{\pi} f(u)\cdot g[\text{pv}(x-u)] du, &&

\big(\text{and }\underbrace{\text{pv}(x) \ \stackrel{\mathrm{def}}{=} \text{Arg}\left(e^{ix}\right)

&= \int_{-\pi}^{\pi} f(u)\cdot g(x-u)\, du, &&\scriptstyle \text{when g(x) is 2}\pi\text{-periodic.}\\

&= \int_{2\pi} f(u)\cdot g(x-u)\, du, &&\scriptstyle \text{when both functions are 2}\pi\text{-periodic, and the integral is over any 2}\pi\text{ interval.}

\end{align}</math>