Let f∊L2π be a real-valued even function with its Fourier series , and let Sn(f,x) be the nth partial sum of the Fourier series, n≧1. The classical result says that if the nonnegative sequence {an} is decreasing and , then if and only if . Later, the monotonicity condition set on {an} 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.
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