We obtain upper bounds for the values of the best bilinear approximations in the Lebesgue spaces of periodic functions of many variables from the Besov-type classes. In special cases, it is shown that these bounds are order exact.
Similar content being viewed by others
References
O. V. Besov, “On a family of functional spaces in connection with embedding and continuity theorems,” Tr. Mat. Inst. Akad. Nauk SSSR, 60, 42–81 (1961).
S. M. Nikol’skii, “Inequalities for entire functions of finite degree and their applications in the theory differentiable functions of many variables,” Tr. Mat. Inst. Akad. Nauk SSSR, 38, 244–278 (1951).
T. I. Amanov, “Representation and embedding theorems for the functional spaces S (r) p,θ B(ℝ n ) and S (r) p,θ ∗ B,” Tr. Mat. Inst. Akad. Nauk SSSR, 77, 5–34 (1965).
P. I. Lizorkin and S. M. Nikol’skii, “Spaces of functions of mixed smoothness from the decomposition point of view” Tr. Mat. Inst. Akad. Nauk SSSR, 187, 143–161 (1989).
A. S. Romanyuk and V. S. Romanyuk, “Best bilinear approximations of functions from Nikol’skii–Besov classes,” Ukr. Mat. Zh., 64, No. 5, 685–697 (2012); English translation: Ukr. Math. J., 64, No. 5, 781–796 (2012).
V. N. Temlyakov, “Approximation of periodic functions of many variables by combinations of functions depending on smaller numbers of variables,” Tr. Mat. Inst. Akad. Nauk SSSR, 173, 243–252 (1986).
V. N. Temlyakov, “Estimates for the best bilinear approximations of periodic functions,” Tr. Mat. Inst. Akad. Nauk SSSR, 181, 250–267 (1988).
V. N. Temlyakov, Approximation of Periodic Functions, Nova Science, New York (1993).
S. M. Nikol’skii, Approximation of Functions of Many Variables and Embedding Theorems [in Russian], Nauka, Moscow (1969).
A. S. Romanyuk, “Bilinear and trigonometric approximations of the Besov classes periodic functions of many variables,” Izv. Ros. Akad. Nauk, Ser. Mat., 70, No. 2, 69–98 (2006).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 12, pp. 1681–1699, December, 2013.
Rights and permissions
About this article
Cite this article
Romanyuk, A.S., Romanyuk, V.S. Best Bilinear Approximations for the Classes of Functions of Many Variables. Ukr Math J 65, 1862–1882 (2014). https://doi.org/10.1007/s11253-014-0903-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-014-0903-1