Integrability and Convergence of Trigonometric Series and Fourier Transforms

A general overview is given for whether a trigonometric series is the Fourier series of an integrable function and when such a series can be approximated in \(L^1\) by its partial sums. This is a necessary preliminary for the attempt to generalize the problem of \(L^1\) convergence of trigonometric series to the non-periodic case, which is the main instance of the present study.