Authors:
R. J. Le Faculty of Science, Ningbo University, Ningbo, Zhejiang, 315211, China

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S. P. Zhou Institute of Mathematics, Zhejiang Sci-Tech University, Hangzhou, Zhejiang, 310018, China

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Abstract

By employing new ideas and techniques, we will refigure out the whole frame of L1-approximation. First, except generalizing the coefficients from monotonicity to a wider condition, Logarithm Rest Bounded Variation condition, we will also drop the prior requirement fL2π but directly consider the sine or cosine series. Secondly, to achieve nontrivial generalizations in complex spaces, we use a one-sided condition with some kind of balance conditions. In addition, a conjecture raised in [9] is disproved in Section 3.

  • [1] Kórus, P. 2010 Remarks on the uniform and L1-convergence of trigonometric series Acta Math. Hungar. 128 369380 .

  • [2] Le, R. J., Zhou, S. P. 2007 On L1 convergence of Fourier series of complex valued functions Studia Sci. Math. Hungar. 44 3547.

  • [3] Leindler, L. 2001 On the uniform convergence and boundedness of a certain class of sine series Anal. Math. 27 279285 .

  • [4] Tikhonov, S. 2008 On L1-convergence of Fourier series J. Math. Anal. Appl. 347 416427 .

  • [5] Xie, T. F., Zhou, S. P. 1996 L1-approximation of Fourier series of complex valued functions Proc. Royal Soc. Edinburgh 126A 343353 .

  • [6] Yu, D. S., Zhou, P., Zhou, S. P. 2009 On L1-convergence of Fourier series under the MVBV condition Canad. Math. Bull. 52 627636 .

  • [7] Zhou, G. Z. 1998 Some remarks on L1-approximation J. Hangzhou Univ. Nat. Ed. 25 1925 (in Chinese).

  • [8] Zhou, S. P. 2010 What condition can correctly generalize monotonicity in L1-convergence of sine series? Science China Math., Chinese Ed. 40 801812 (in Chinese).

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  • [9] Wang, M. Z., Zhou, S. P. 2010 Applications of MVBV condition in L1 integrability Acta Math. Hungar. 129 7080 .

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Acta Mathematica Hungarica
Language English
Size B5
Year of
Foundation
1950
Volumes
per Year
3
Issues
per Year
6
Founder Magyar Tudományos Akadémia  
Founder's
Address
H-1051 Budapest, Hungary, Széchenyi István tér 9.
Publisher Akadémiai Kiadó
Springer Nature Switzerland AG
Publisher's
Address
H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
CH-6330 Cham, Switzerland Gewerbestrasse 11.
Responsible
Publisher
Chief Executive Officer, Akadémiai Kiadó
ISSN 0236-5294 (Print)
ISSN 1588-2632 (Online)

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