Abstract: In this paper, the exact values of the extreme characteristic of a special kind, which connects the best polynomial approximation of analytical functions in a unit circle \(f\in B_{2}^{(r)}\) \((r\in\mathbb{Z}_{+})\) with modulus of continuity of \(m\)-th order \(\omega_{m}(z^{r}f^{(r)},u)_{B_2}\) of the \(r\)-th derivative \(f^{(r)}\) belonging to the Bergman space \(B_2\), are found. In this case, the extreme characteristic contains this modulus of continuity not only under the sign of the integral, but also outside the integral. The statement proved in Theorem 1 generalizes a well-known result by S. B. Vakarchuk and A. N. Shitova [17], obtained for the classical
modulus of continuity of \(m\)-th order \(\omega_{m}(f^{(r)},t) _{L_2}\) of periodic functions \(f\in L_{2}[0,2\pi]\) with derivative \(f^{(r)}\in L_{2}[0,2\pi]\), to the case of the modulus of continuity \(\omega_{m}(z ^{r} f^{(r)},u) _{B_2}\) of analytic functions in the unit circle with derivative \(f^{(r)}\in B_2\). The obtained result provides the opportunity to compute the exact values of the Bernstein, Kolmogorov, linear, Gelfand, and projection \(n\)-widths of certain classes of functions in the Bergman space \(B_2\).
Keywords: root-mean-square approximation of analytic functions in the disk, Bergman space, \(n\)-width
For citation: Palavonov, Q. Q. and Khuromonov, Kh. M. The Root-Mean-Square Approximations of Functions in Bergman Space and the Value of the Widths of Some Classes of Functions, Vladikavkaz Math. J., 2026, vol. 28, no. 3, pp.108-120. (in Russian). DOI 10.46698/v5873-0367-2681-e
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