A remark on a condition raised by Tikhonov: An example in L 1 convergence of fourier series

Abstract

By constructing an example, the present note will show that the condition GM raised by Tikhonov [2] to study L ∞ convergence case cannot keep the classical theorem working in L 1 convergence case.In the past ninety years, many classical results in Fourier analysis have been generalized by weakening the condition imposed on the coefficients of trigonometric series from MS (monotone non-increasing sequences) to QMS (quasi-monotone sequences), RVQMS (O-regularly varying quasimonotone sequences), RBVS (rest bounded variation sequences), GBVS (group bounded variation sequences), NBVS (non-onesided bounded variation sequences) and, finally, the MVBVS (mean value bounded variation sequences). The history of development of these sequences (or in other words, these conditions) can be found in reference [5] or [6].[5] recently, and applied to uniform convergence [5], L 1 -convergence [4] and L p integrability [3] of trigonometric series. Among the important classical results for trigonometric series, the first and the most important one is the classical Chaundy-Jolliffe's theorem [1] for uniform convergence of sine series in 1916, which states that if {a n } is a nonnegative and non-increasing sequence ({a n } ∈ MS) with lim n→∞ a n = 0, then a necessary and sufficient condition for the uniform convergence of series ∞ n=1 a n sin nx is lim n→∞ na n = 0.The definition of MVBVS can be read as follows: