On the best polynomial approximation of functions in the Hardy space 𝐻𝑞,𝑅, (1 ⩽ 𝑞 ⩽ ∞, 𝑅 ⩾ 1)
https://doi.org/10.22405/2226-8383-2023-24-1-182-193
Abstract
Exact inequalities are found between the best polynomial approximation of functions analytics in the disk 𝑈𝑅 :={︀𝑧 ∈ C, |𝑧| < 𝑅}︀, 𝑅 ⩾ 1 and the averaged modulus of continuity angular boundary values of the 𝑚th order derivatives. For the class 𝑊(𝑚) 𝑞,𝑅 (𝑚 ∈ Z+, 1 ⩽ 𝑞 ⩽ ∞, 𝑅 ⩾ 1) of functions 𝑓 ∈ 𝐻(𝑚) 𝑞,𝑅 whose 𝑚-order derivatives 𝑓(𝑚) belong to the Hardy space 𝐻𝑞,𝑅 and satisfy the condition ‖𝑓(𝑚)‖𝑞,𝑅 ⩽ 1, the exact values of the upper bounds of the best approximations are calculated. Moreover, for the class 𝑊(𝑚) 𝑞,𝑅 (Φ), consisting of all functions 𝑓 ∈ 𝐻(𝑚) 𝑞,𝑅 , for which any 𝑘 ∈ N, 𝑚 ∈ Z+, 𝑘 > 𝑚 the averaged moduli of continuity of the boundary values of the 𝑚th order derivative 𝑓(𝑚), dominated in the system of points {𝜋/𝑘}𝑘∈N
by the given function Φ, satisfy the condition
$$∫︁(0, 𝜋/𝑘) 𝜔(︀𝑓^(𝑚), 𝑡)︀_(𝑞,𝑅) 𝑑𝑡 ⩽ Φ(𝜋/𝑘),$$
the exact values of the Kolmogorov and Bernstein 𝑛-widths are calculated in the norm of the space 𝐻𝑞 (1 ⩽ 𝑞 ⩽ ∞).
The results obtained generalize some results of L.V.Taikov on classes of analytic functions in a circle of radius 𝑅 ⩾ 1.
About the Authors
Mirgand Shabozovich Shabozov
Tajik National University
Tajikistan
Tajikistan
doctor of physical and mathematical sciences, professor
Gulzorkhon Amirshoevich Yusupov
Tajik State S.Aini Pedagogical University
Tajikistan
Tajikistan
doctor of physical and mathematical sciences, associate
professor
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Review
For citations:
Shabozov M.Sh., Yusupov G.A. On the best polynomial approximation of functions in the Hardy space 𝐻𝑞,𝑅, (1 ⩽ 𝑞 ⩽ ∞, 𝑅 ⩾ 1). Chebyshevskii Sbornik. 2023;24(1):182-193. (In Russ.) https://doi.org/10.22405/2226-8383-2023-24-1-182-193
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