On L1-Convergence of Fourier Series under the MVBV Condition

Let f ∈ L2π be a real-valued even function with its Fourier series , and let Sn( f , x), n ≥ 1, be the n-th partial sum of the Fourier series. It is well known that if the nonnegative sequence ﹛an﹜ is decreasing and limn→∞an = 0, thenWe weaken the monotone condition in this classical result to the so-called mean value bounded variation (MVBV) condition. The generalization of the above classical result in real-valued function space is presented as a special case of the main result in this paper, which gives the L1-convergence of a function f ∈ L2π in complex space. We also give results on L1-approximation of a function f ∈ L2π under the MVBV condition.