Abstract
We obtain the exact-by-order estimates of the approximation of functions from the classes \( {S}_{p,\theta}^{\varOmega } \) B(ℝd) in the space Lq(ℝd), 1 < p < q < ∞, by entire functions of the exponential type with supports of their Fourier transforms in sets generated by the level surfaces of a function Ω.
Similar content being viewed by others
References
S. A. Stasyuk and S. Ya. Yanchenko, “Approximation of functions from Nikol’skii–Besov type classes of generalized mixed smoothness,” Anal. Math., 41, 311–334 (2015).
S. Ya. Yanchenko, “Ordinal estimates of approximative characteristics of functions from the classes \( {S}_{p,\theta}^{\varOmega } \) (ℝd) with given majorant of mixed moduli of continuity in a uniform metric,” Ukr. Mat. Zh., 68, No. 12, 1705–1714 (2016).
N. K. Bari and S. B. Stechkin, “The best approximations and differential properties of two conjugate functions,” Trudy Mosk. Mat. Obshch., 5, 483–522 (1956).
S. M. Nikol’skii, “Functions with dominant mixed derivative satisfying the multiple Hölder condition,” Sibir. Mat. Zh., 4, No. 6, 1342–1364 (1963).
T. I. Amanov, “Representation and imbedding theorems for the functional spaces \( {S}_{p,\theta}^{\varOmega } \) B(ℝn) and \( {S}_{p,\theta}^{\varOmega } \) (0 ≤ xj ≤ 2 π; j = 1, . . . , n),” Trudy Mat. Inst. Akad. Nauk SSSR, 77, 5–34 (1965).
T. I. Amanov, Spaces of Differentiable Functions with Dominating Mixed Derivative [in Russian], Nauka, Alma-Ata (1976).
Wang Heping and Sun Yongsheng, “Approximation of multivariate functions with a certain mixed smoothness by entire functions,” Northeast. Math. J., 11, No. 4, 454–466 (1995).
Heping Wang, “Representation and approximation of multivariate function with bounded mixed smoothness by hyperbolic wavelets,” J. Math. Anal. Appl., 291, 698–715 (2004).
V. N. Temlyakov, “Approximation of functions with bounded mixed derivative,” Trudy Mat. Inst. Akad. Nauk SSSR, 178, 1–112 (1986).
A. S. Romanyuk, “Approximative Characteristics of the Classes of Periodic Functions of Many Variables” [in Russian], Institute of Mathematics of the NAS of Ukraine, Kiev (2012).
N. N. Pustovoitov, “The representation and approximation of periodic functions of many variables with given mixed modulus of continuity,” Anal. Math., 20, 35–48 (1994).
N. N. Pustovoitov, “On the Kolmogorov widths of the classes of functions with given mixed modulus of continuity,” Anal. Math., 38, No. 1, 41–64 (2012).
Sun Yongsheng and Wang Heping, “Representation and approximation of multivariate periodic functions with bounded mixed moduli of smoothness,” Trudy Mat. Inst. Ross. Akad. Nauk, 219, 356–377 (1997).
Liqin Duan, “The best m-term approximations on generalized Besov classes \( M{B}_{q,\theta}^{\varOmega } \) with regard to orthogonal dictionaries,” J. of Approx. Theory, 162, 1964-1981 (2010).
P. I. Lizorkin and S. M. Nikol’skii, “Function spaces of mixed smoothness from the decomposition point of view,” Trudy Mat. Inst. Akad. Nauk SSSR, 187, 143–161 (1989).
P. I. Lizorkin, “Generalized Liouville differentiation and the multiplier method in the theory of embeddings of the classes of differentiable functions,” Trudy Mat. Inst. Akad. Nauk SSSR, 105, 89–167 (1969).
Din’ Zung, “Approximation of functions of many variables on a torus by trigonometric polynomials,” Mat. Sbornik, 131(173), No. 2(10), 251–271 (1986).
A. S. Romanyuk, “Approximation of the Besov classes of periodic functions of many variables in the space Lq,” Ukr. Mat. Zh., 43, No. 10, 1398–1408 (1991).
A. S. Romanyuk, “On the approximation of the classes of periodic functions of many variables,” Ukr. Mat. Zh., 44, No. 5, 662–672 (1992).
N. N. Pustovoitov, “Approximation of multidimensional functions with given majorant of mixed moduli of continuity,” Mat. Zametki, 65, No. 1, 107–117 (1999).
S. A. Stasyuk, “The best approximations of periodic functions of many variables from the classes \( {B}_{p,\theta}^{\varOmega } \),” Mat. Zametki, 87, No. 1, 108–121 (2010).
S. A. Stasyuk, “Approximation by Fourier sums and Kolmogorov widths of the classes \( M{B}_{p,\theta}^{\varOmega } \) of periodic functions of several variables,” Trudy Inst. Mat. Mekh. UrO Ross. Akad. Nauk, 20, No. 1, 247–257 (2014).
Sh. A. Balgimbaeva and T. I. Smirnov, “Estimates of the Fourier widths of the classes of periodic functions with mixed modulus of smoothness,” Trudy Inst. Mat. Mekh. UrO Ross. Akad. Nauk, 21, No. 4, 78–94 (2015).
D. Ding, V. N. Temlyakov, and T. Ullrich, Hyperbolic Cross Approximation, arXiv:1601.03978v3 [math.NA] 21 Apr 2017.
S. M. Nikol’skii, Approximation of Functions of Many Variables and Embedding Theorems [in Russian], Nauka, Moscow, 1969.
P. I. Lizorkin, “A Littlewood–Paley type theorem for multiple Fourier integrals,” Trudy Mat. Inst. Akad. Nauk SSSR, 89, 214–230 (1967).
S. Ya. Yanchenko, “Approximation of the classes \( {B}_{p,\theta}^{\varOmega } \) of functions of many variables in the space L q(ℝd),” Ukr. Mat. Zh., 62, No. 1, 123–135 (2010).
G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities, Cambridge Univ. Press, Cambridge (1934).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 15, No. 1, pp. 132–148 January–March, 2018.
Rights and permissions
About this article
Cite this article
Yanchenko, S.Y., Stasyuk, S.A. Approximative characteristics of functions from the classes \( {S}_{p,\theta}^{\varOmega } \) B(ℝd) with a given majorant of mixed moduli of continuity. J Math Sci 235, 103–115 (2018). https://doi.org/10.1007/s10958-018-4062-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-018-4062-z