On L 1-convergence of Fourier series of complex valued functions under the GM7 condition

Let f??L _2?? be a real-valued even function with its Fourier series $\frac {a_{0}}{2}+\sum\limits_{n=1}^{\infty}{a_{n}}\cos nx$ , and let S _n (f,x) be the nth partial sum of the Fourier series, n?R1. The classical result says that if the nonnegative sequence {a _n } is decreasing and $\lim\limits_{n\to\infty} a_{n} =0$ , then $\lim\limits_{n\to\infty} \|{f-S_{n}(f)}\|_{L}=0$ if and only if $\lim\limits_{n\to\infty} a_{n}\log n=0$ . Later, the monotonicity condition set on {a _n } is essentially generalized to MVBV (Mean Value Bounded Variation) condition. Very recently, Kórus further generalized the condition in the classical result to the so-called GM₇ condition in real space. In this paper, we give a complete generalization to the complex space.