zygmund space
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2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
H. K. Nigam ◽  
Mohammad Mursaleen ◽  
Supriya Rani

AbstractIn this paper we review some of the previous work done by the earlier authors (Singh et al. in J. Inequal. Appl. 2017:101, 2017; Lal and Shireen in Bull. Math. Anal. Appl. 5(4):1–13, 2013), etc., on error approximation of a function g in the generalized Zygmund space and resolve the issue of these works. We also determine the best error approximation of the functions g and $g^{\prime }$ g ′ , where $g^{\prime }$ g ′ is a derived function of a 2π-periodic function g, in the generalized Zygmund class $X_{z}^{(\eta )}$ X z ( η ) , $z\geq 1$ z ≥ 1 , using matrix-Cesàro $(TC^{\delta })$ ( T C δ ) means of its Fourier series and its derived Fourier series, respectively. Theorem 2.1 of the present paper generalizes eight earlier results, which become its particular cases. Thus, the results of (Dhakal in Int. Math. Forum 5(35):1729–1735, 2010; Dhakal in Int. J. Eng. Technol. 2(3):1–15, 2013; Nigam in Surv. Math. Appl. 5:113–122, 2010; Nigam in Commun. Appl. Anal. 14(4):607–614, 2010; Nigam and Sharma in Kyungpook Math. J. 50:545–556, 2010; Nigam and Sharma in Int. J. Pure Appl. Math. 70(6):775–784, 2011; Kushwaha and Dhakal in Nepal J. Sci. Technol. 14(2):117–122, 2013; Shrivastava et al. in IOSR J. Math. 10(1 Ver. I):39–41, 2014) become particular cases of our Theorem 2.1. Several corollaries are also deduced from our Theorem 2.1.


2020 ◽  
Vol 27 (4) ◽  
pp. 569-575
Author(s):  
Alexander Kharazishvili
Keyword(s):  

AbstractFor certain families of topologies, the existence of a common Sierpiński–Zygmund function (of a common Sierpiński–Zygmund function in the strong sense) is established. In this connection, the notion of a Sierpiński–Zygmund space (of a Sierpiński–Zygmund space in the strong sense) is introduced and examined. The behavior of such spaces under some standard topological operations is considered.


2020 ◽  
Vol 101 (3) ◽  
pp. 466-476
Author(s):  
MUNIRAH ALJUAID ◽  
FLAVIA COLONNA

In this paper we study a class ${\mathcal{Z}}_{H}$ of harmonic mappings on the open unit disk $\mathbb{D}$ in the complex plane that is an extension of the classical (analytic) Zygmund space. We extend to the elements of this class a characterisation that is valid in the analytic case. We also provide a similar result for a closed separable subspace of ${\mathcal{Z}}_{H}$ which we call the little harmonic Zygmund space.


2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Munirah Aljuaid ◽  
Flavia Colonna

We study the composition operators on Banach spaces of harmonic mappings that extend several well-known Banach spaces of analytic functions on the open unit disk in the complex plane, including the α-Bloch spaces, the growth spaces, the Zygmund space, the analytic Besov spaces, and the space BMOA.


2019 ◽  
Vol 52 (1) ◽  
pp. 97-104
Author(s):  
Tat-Leung Yee ◽  
Kwok-Pun Ho

AbstractWe generalize the Zygmund inequality for the conjugate function to the Morrey type spaces defined on the unit circle T. We obtain this extended Zygmund inequality by introducing the Morrey-Zygmund space on T.


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