On the integrability and $L^ 1$-convergence of complex trigonometric series

Abstract:

We prove that if a weakly even sequence $\{ {c_k}:k = 0, \pm 1, \ldots \}$ of complex numbers is such that for some $p > 1$ we have \[ \sum \limits _{m = 1}^\infty {{2^{m/q}}} {\left ( {\sum \limits _{k = {2^{m - 1}}}^{{2^m} - 1} {{{\left | {\Delta \left ( {{c_k} + {c_{ - k}}} \right )} \right |}^p}} } \right )^{1/p}} < \infty ,\frac {1}{p} + \frac {1}{q} = 1,\] then the symmetric partial sums of the trigonometric series $( * )\sum \nolimits _{k = - \infty }^\infty {{c_k}{e^{ikx}}}$ converge pointwise, except possibly at $x = 0(\operatorname {mod} 2\pi )$, to a Lebesgue integrable function, $( * )$ is the Fourier series of its sum, and series $( * )$ converges in ${L^1}( - \pi ,\pi )$-norm if and only if ${\lim _{|k| \to \infty }}{c_k}\ln |k| = 0$. In addition, we present new proofs of the theorems by J. Fournier and W. Self [6] and by ČC. V. Stanojević and V. B. Stanojević [10].