Addendum to “Trigonometric series of Nikol’skii classes”

Tikhonov, Sergey

doi:10.1007/s10474-008-7074-1

Acta Math Hung

2008

DOI: 10.1007/s10474-008-7074-1

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Addendum to “Trigonometric series of Nikol’skii classes”

Sergey Tikhonov

Abstract: We describe the functions from Nikol'skii class in terms of behavior of their Fourier coecients. Results for series with general monotone coecients are presented. The problem of strong approximation of Fourier series is also studied.

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

Zhou

2010

Acta Math Hung

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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:

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

Zhou

2010

Acta Math Hung

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Addendum to “Trigonometric series of Nikol’skii classes”

Abstract: We describe the functions from Nikol'skii class in terms of behavior of their Fourier coecients. Results for series with general monotone coecients are presented. The problem of strong approximation of Fourier series is also studied.

Cited by 1 publication

References 8 publications

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

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

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