Approximation of functions by linear summation methods in the Orlicz-type spaces
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Approximation of functions by linear summation methods in the Orlicz-type spaces Stanislav Chaichenko, Viktor Savchuk, Andrii Shidlich (Presented by F. Abdullaev) Abstract. Approximative properties of linear summation methods of Fourier series are considered in the Orlicz-type spaces SM . In particular, in terms of approximations by such methods, constructive characteristics are obtained for the classes of functions whose moduli of smoothness do not exceed a certain majorant.
Keywords. Linear summation method, modulus of smoothness, direct approximation theorem, inverse approximation theorem, Orlicz-type spaces.
1.
Introduction
Linear methods (or processes) of summation of Fourier series are an important object of research in approximation theory. In particular, this is due to the fact that most of these methods naturally generate the corresponding aggregate of approximation. These topics are well studied in classical functional spaces such as the Lebesgue and Hilbert ones, the spaces of continuous functions, etc. However, there are relatively a few papers devoted to similar topics in Banach spaces of the Orlicz type. Particularly, this concerns the direct and inverse theorems of approximation by linear summation methods. In the present paper, the approximative properties of linear summation methods of Fourier series are studied in the Orlicz-type spaces SM . The spaces SM are defined in the following way. The Orlicz function M (t), t ≥ 0 is a non-decreasing convex function and is such that M (0) = 0 and M (t) → ∞ as t → ∞. Let SM be the space of all 2π-periodic Lebesgue summable functions f (f ∈ L1 ) such that the following quantity (which is also called the Luxemburg norm of f ) is finite: ∥f ∥M := ∥{fb(k)}k∈Z ∥l
{ M (Z)
= inf a > 0 :
∑
} b M (|f (k)|/a) ≤ 1 ,
(1.1)
k∈Z
∫ 2π where fb(k) := [f ]b(k) = (2π)−1 0 f (t)e−ikt dt, k ∈ Z, are the Fourier coefficients of f . Functions f ∈ L1 and g ∈ L1 are equivalent in the space SM when ∥f − g∥M = 0. The spaces SM defined in this way are Banach spaces. They were considered in [6]. In particular, the direct and inverse approximation theorems in terms of the best approximations of functions and moduli of fractional smoothness were proved for the spaces SM in [6]. Translated from Ukrains’ki˘ı Matematychny˘ı Visnyk, Vol. 17, No. 2, pp. 152–170 April–June, 2020. Original article submitted February 02, 2020 This work is partially supported by the grant H2020-MSCA-RISE-2014 (project number 645672), President’s of Ukraine grant for competitive projects (project number F84/177-2019), and a grant of the NAS of Ukraine to research groups of young scientists (project number 04-02/2019). c 2020 Springer Science+Business Media, LLC 1072 – 3374/20/2495–0705 ⃝
705
In the case M (t) = tp , p ≥ 1, the spaces SM coincide with the well-known spaces S p [18] of functions f ∈ L1 with finite norm )1/p (∑ ∥f ∥S p = ∥{fb(k)}k∈Z ∥lp (Z) = |fb(k)|p . k∈Z
In S p , the approximative properties of linear summation methods of Fourier series were studied in [16, 17]. The purpose of this paper
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