On the $L_1$-convergence of Fourier series
Since the trigonometric Fourier series of an integrable function does not necessarily converge to the function in the mean, several additional conditions have been devised to guarantee the convergence. For instance, sufficient conditions can be constructed by using the Fourier coefficients or the integral modulus of the corresponding function. In this paper we give a Hardy-Karamata type Tauberian condition on the Fourier coefficients and prove that it implies the convergence of the Fourier series in integral norm, almost everywhere, and if the function itself is in the real Hardy space, then also in the Hardy norm. We also compare it to the previously known conditions.