A generalization of the monotonicity condition and applications

Yu, Dansheng; Zhou, S. P.

doi:10.1007/s10474-007-5253-0

Cited by 24 publications

(33 citation statements)

References 10 publications

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“…A particular case of the notion of MVBVS was earlier introduced in [9] as follows. A sequence {a k } ⊂ R + is said to belong to the class NBVS (= non-onesided bounded variation sequences) if there exists a constant C, depending only on {a k }, such that The following theorem was also formulated in [11] and supported with a counterexample.…”

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confidence: 99%

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On the uniform convergence of double sine series

Kórus¹,

Móricz²

2009

Studia Math.

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Abstract. Let a single sine series ( * ) P ∞ k=1 a k sin kx be given with nonnegative coefficients {a k }. If {a k } is a "mean value bounded variation sequence" (briefly, MVBVS), then a necessary and sufficient condition for the uniform convergence of series ( * ) is that ka k → 0 as k → ∞. The class MVBVS includes all sequences monotonically decreasing to zero. These results are due to S. P. Zhou, P. Zhou and D. S. Yu. In this paper we extend them from single to double sine series ( * * ) P ∞ k=1 P ∞ l=1 c kl sin kx sin ly, even with complex coefficients {c kl }. We also give a uniform boundedness test for the rectangular partial sums of series ( * * ), and slightly improve the results on single sine series.

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confidence: 99%

“…An analysis of the proofs in [9] and [11] reveals that the sufficiency part in Theorem B as well as Theorem C remain valid if we consider sequences {c k : k = 1, 2, . .…”

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confidence: 99%

On the uniform convergence of double sine series

Kórus¹,

Móricz²

2009

Studia Math.

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“…The monotonic condition in the Chaundy-Jollif Theorem is then extended to {a n } ∈ GBVS. Later, Yu and Zhou [25] introduced further the non-onesided bounded variation condition. A nonnegative sequence A = {a n } is said to be a non-onesided bounded variation sequence ({a n } ∈ NBVS) if 2n k=n |a k − a k+1 | ≤ C(A)(a n + a 2n ) holds for some constant C(A) and all n = 1, 2, · · · .…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…Previous results concerning the generalization of ChaundyJollif Theorem to complex space can be found in [7], [23], and [25], etc. In this section we establish the following Theorem 10 Let C = {c n } be a complex sequence satisfying…”

Section: Results In Complex Spacementioning

confidence: 99%

Ultimate generalization to monotonicity for uniform convergence of trigonometric series

Zhou

2010

Sci. China Math.

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Chaundy and Jolliffe [4] proved that if {a n } is a non-increasing (monotonic) real sequence with lim n→∞ a n = 0, then a necessary and sufficient condition for the uniform convergence of the series ∞ n=1 a n sin nx is lim n→∞ na n = 0. We generalize (or weaken) the monotonic condition on the coefficient sequence {a n } in this classical result to the so-called mean value bounded variation condition and prove that the generalized condition cannot be weakened further. We also establish an analogue to the generalized Chaundy and Jolliffe theorem in the complex space.2000 Mathematics Subject Classification. 42A20 42A32.

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“…Several classes of sequences have been introduced to generalize Theorem 1 (see [10], [9], [2], [6]). These classes are larger than the class of monotone sequences and contain sequences of complex numbers as well.…”

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confidence: 99%

On the uniform convergence of double sine series

Duzinkiewicz

Szal

2018

Colloq. Math.

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The fundamental theorem in the theory of the uniform convergence of sine series is due to Chaundy and Jolliffe from 1916 (see [1]). Several authors gave conditions for this problem supposing that coefficients are monotone, non-negative or more recently, general monotone (see [8], [6] and [2], for example). There are also results for the regular convergence of double sine series to by uniform in case the coefficients are monotone or general monotone double sequences. In this article we give new sufficient conditions for the uniformity of the regular convergence of double sine series, which are necessary as well in case the coefficients are non-negative. We shall generalize those results defining a new class of double sequences for the coefficients.Mathematics subject classification number: 42A20, 42A32, 42B99.

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A generalization of the monotonicity condition and applications

Abstract: We introduce a new class of sequences called NBVS to generalize GBVS, essentially extending monotonicity from one sided to two sided, while some important classical results keep true.

Cited by 24 publications

References 10 publications

On the uniform convergence of double sine series

On the uniform convergence of double sine series

Ultimate generalization to monotonicity for uniform convergence of trigonometric series

On the uniform convergence of double sine series

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