Abstract：Xie and Zhou establishes some nice results fo r L1-Approximation of Fourier series of complex valued functions w ith O-regularly varying quasimonotone coefficients.Exactly , one result says that ‖ f - S n(f)‖ = O(ψn) En(f)L = O(ψn)and f(n)log |n |=O(ψ|n|), where {ψn}is a decreasing null sequence satisfying ψn =O(ψ2 n).It is naturalto ask if the conditio n En(f)L = O(ψn)can be removed under some conditions? The persent paper will investigate this question .
