Abstract
We study the approximative properties of L q -greedy algorithms with respect to the wellknown system U d of shifts of Dirichlet kernels on the Nikol′skiĭ-Besov classes \(SB_{p\theta }^r (\mathbb{T}^d )\) and the Lizorkin-Triebel classes \(SF_{p\theta }^r (\mathbb{T}^d )\) of functions of mixed smoothness.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Temlyakov V. N., “Greedy approximation,” Acta Numer., 17, 235–409 (2008).
Temlyakov V. N., “Greedy algorithms with regard to multivariate systems with special structure,” Constr. Approx., 16, No. 3, 399–425 (2000).
Amanov T. I., “Representation and embedding theorems for function spaces Sp,θ (r)B(Rn) and Sp*,θ (r)B(Rn) (0 ≦ xj ≦ 2π j = 1, …, n),” Trudy Mat. Inst. Steklov., 77, 5–34 (1965).
Lizorkin P. I. and Nikol′skiĭ S. M., “Function spaces of mixed smoothness from decompositional point of view,” Proc. Steklov Inst. Math., 187, 163–184 (1990).
Nikol′skiĭ S. M., “A representation theorem for a class of differentiable functions of several variables in terms of entire functions of exponential type,” Dokl. Akad. Nauk SSSR, 150, No. 3, 484–487 (1963).
Nikol′skiĭ S. M., “Functions with dominant mixed derivative, satisfying a multiple H¨older condition,” Sibirsk. Mat. Zh., 4, No. 6, 1342–1364 (1963).
Babenko K. I., “On approximation of periodic functions of several variables by trigonometric polynomials,” Dokl. Akad. Nauk SSSR, 132, No. 2, 247–250 (1960).
Babenko K. I., “On approximation of one class of periodic functions of several variables by trigonometric polynomials,” Dokl. Akad. Nauk SSSR, 132, No. 5, 982–985 (1960).
Nikol′skiĭ S. M., “Boundary properties of differentiable functions of several variables,” Dokl. Akad. Nauk SSSR, 146, No. 3, 542–545 (1962).
Nikol′skiĭ S. M., “Stable boundary-value problems of a differentiable function of several variables,” Mat. Sb., 61, No. 2, 224–252 (1963).
Temlyakov V. N., “Approximations of functions with bounded mixed derivative,” Proc. Steklov Inst. Math., 178, 1–121 (1989).
Temlyakov V. N., Approximation of Periodic Functions, Nova Sci. Publ., New York (1993).
Temlyakov V. N., “Nonlinear methods of approximation,” Found. Comp. Math., 3, No. 1, 33–107 (2003).
Amanov T. I., Spaces of Differentiable Functions with Dominating Mixed Derivative [in Russian], Nauka, Alma-Ata (1976).
Schmeisser H.-J. and Triebel H., Topics in Fourier Analysis and Function Spaces, Wiley, Chichester (1987).
Schmeisser H.-J., “Recent developments in the theory of function spaces with dominating mixed smoothness,” in: Nonlinear Analysis, Function Spaces and Applications: Proc. Spring School held in Prague, May 30–June 6, 2006, Czech Acad. Sci., Math. Inst., Prague, 2007, V. 8, pp. 145–204.
Nikol′skiĭ S. M., Approximation of Functions of Several Variables and Imbedding Theorems, Springer, Berlin (1975).
Dung D., “Continuous algorithms in n-term approximation and non-linear widths,” J. Approx. Theory, 102, No. 2, 217–242 (2000).
Bazarkhanov D. B., “Nonlinear approximations of classes of periodic functions of many variables,” Proc. Steklov Inst. Math., 284, 2–31 (2014).
Kashin B. S. and Saakyan A. A., Orthogonal Series, Amer. Math. Soc., Providence (1999).
Wojtaszczyk P., “On unconditional polynomial bases in Lp and Bergman spaces,” Constr. Approx., 13, No. 1, 1–15 (1997).
Author information
Authors and Affiliations
Corresponding author
Additional information
The author was supported by Grants 0740/GF and 0245/GF3 of the Ministry of Education and Science of the Republic of Kazakhstan.
Almaty. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 56, No. 2, pp. 322–337, March–April, 2015.
Rights and permissions
About this article
Cite this article
Balgimbayeva, S.A. Nonlinear approximation of function spaces of mixed smoothness. Sib Math J 56, 262–274 (2015). https://doi.org/10.1134/S0037446615020068
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446615020068