Mean-squared approximation of some classes of complex variable functions by Fourier series in the weighted Bergman space 𝐵_2,𝛾
https://doi.org/10.22405/2226-8383-2022-23-1-167-182
Abstract
The article considers extremal problems of mean-square approximation of functions of a complex variable, regular in the domain D ⊂ C, by Fourier series orthogonal in the system of functions {𝜙_𝑘(𝑧)}∞𝑘=0 in D belonging to the weighted Bergman space 𝐵2,𝛾 with finite norm
$$‖𝑓‖2,𝛾 := ‖𝑓‖𝐵2,𝛾 =(1/2𝜋∫︁∫︁(D) 𝛾(|𝑧|)|𝑓(𝑧)|^2 𝑑𝜎)^(1/2),$$
where 𝛾 := 𝛾(|𝑧|) ≥ 0 is a real integrable function in the domain D, and the integral is understood in the Lebesgue sense, 𝑑𝜎 := 𝑑𝑥𝑑𝑦 is an element of area.
The formulated problem is investigated in more detail in the case when D is the unit disc in the space 𝐵_2,𝛾𝛼,𝛽 , 𝛾_𝛼,𝛽 = |𝑧|^𝛼(1 − |𝑧|)^𝛽, 𝛼, 𝛽 > −1 – Jacobi weight. Sharp Jackson-Stechkintype inequalities that relate the value of the best mean-squared polynomial approximation of 𝑓 ∈ ℬ^(𝑟)_2,𝛾𝛼,𝛽 and the Peetre K -functional were proved. In case when 𝛾𝛼,𝛽 ≡ 1 we will obtain the earlier known results.
About the Authors
Mirgand Shabozovich Shabozov
Tajik National University
Tajikistan
Tajikistan
doctor of physical and mathematical sciences, professor, academician of the National Academy of Sciences of Tajikistan
Mukim Saidusainovich Saidusainov
Tajik National University
Tajikistan
Tajikistan
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Review
For citations:
Shabozov M.Sh., Saidusainov M.S. Mean-squared approximation of some classes of complex variable functions by Fourier series in the weighted Bergman space 𝐵_2,𝛾. Chebyshevskii Sbornik. 2022;23(1):167-182. (In Russ.) https://doi.org/10.22405/2226-8383-2022-23-1-167-182
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