Abstract
In normed spaces of functions analytic in the Jordan domain Ω⊂ℂ, we establish exact order estimates for the Kolmogorov widths of classes of functions that can be represented in Ω by Cauchy-type integrals along Γ = ∂Ω with densities f(·) such that \(f \circ \Psi \in L_{\beta ,p}^\Psi (T)\). Here, Ψ is a conformal mapping of \(C\backslash \overline \Omega \) onto {w: |w| > 1}, and \(L_{\beta ,p}^\Psi (T)\) is a certain subset of infinitely differentiable functions on T = {w: |w| = 1}.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
V. S. Romanyuk, “Estimates of the Kolmogorov widths of classes of analytic functions representable by Cauchy-type integrals. I,” Ukr. Mat. Zh. 53, No. 2, 229–237 (2001).
S. B. Vakarchuk, “On the widths of certain classes of analytic functions. I,” Ukr. Mat. Zh. 44, No. 3, 324–333 (1992).
V. S. Romanyuk, “Weighted approximation in mean of classes of analytic functions by algebraic polynomials and finite-dimensional subspaces,” Ukr. Mat. Zh. 51, No. 5, 645–662 (1999).
V. S. Romanyuk, “Weak asymptotics of the Kolmogorov widths of classes of convolutions of periodic functions,” in: Abstracts of the International Conference “Approximation Theory and Numerical Methods” (Rivne, June 19–21, 1996) [in Ukrainian], Rivne (1996), p. 66.
V. N. Temlyakov, “On estimates for the widths of classes of infinitely differentiable functions,” Mat. Zametki 47, No. 5, 155–157 (1990).
V. N. Temlyakov, “On estimates of the widths of classes of infinitely differentiable functions,” Dokl. Rasshir. Zased. Sem. Inst. Prikl. Mat. im. I. N. Vekua 5, No. 2, 111–114 (1990).
A. K. Kushpel', “Estimates of the Bernshtein widths and their analogs,” Ukr. Mat. Zh. 45, No. 1, 54–59 (1993).
A. I. Stepanets and A. K. Kushpel', Best Approximations and Widths of Classes of Periodic Functions [in Russian], Preprint No. 84.15, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1984).
A. K. Kushpel', Widths of Classes of Smooth Functions in the Space L q [in Russian], Preprint No. 87.44, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1987).
M. Z. Dveirin, “Widths and ε-entropy of classes of functions analytic in the unit circle,” Teor. Funkts., Funkts. Anal. Prilozhen. Issue 3, 32–46 (1975).
V. E. Maiorov, “On the best approximation of the classes W r1 (I s ) in the space Lε(I s ),” Mat. Zametki 19, No. 5, 699–706 (1976).
A. Zygmund, Trigonometric Series Vol. 1, Cambridge University Press, Cambridge (1959).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Romanyuk, V.S. Estimates of the Kolmogorov Widths of Classes of Analytic Functions Representable by Cauchy-Type Integrals. II. Ukrainian Mathematical Journal 53, 395–406 (2001). https://doi.org/10.1023/A:1012340204640
Issue Date:
DOI: https://doi.org/10.1023/A:1012340204640