We establish exact-order estimates for the trigonometric widths of the Nikol’skii–Besov classes of periodic functions of many variables in the Lebesgue space with mixed norm.
Similar content being viewed by others
References
S. M. Nikol’skii, Approximation of Functions of Many Variables and Imbedding Theorems [in Russian], Nauka, Moscow (1977).
O. V. Besov, “Investigation of one family of functional spaces in connection with imbedding and continuation theorems,” Tr. Mat. Inst. Akad. Nauk SSSR, 60, 42–81 (1961).
R. S. Ismagilov, “Widths of the sets in normalized linear spaces and the approximation of functions by trigonometric polynomials,” Usp. Mat. Nauk, 29, No. 3, 161–178 (1974).
Ya. S. Bugrov, “Approximation of a class of functions with dominating mixed derivative,” Mat. Sb., 64(106), No. 3, 410–418 (1964).
É. S. Belinskii, “Approximation of periodic functions by a ‘floating’ system of exponents and trigonometric widths,” in: Investigation in the Theory of Functions of Many Real Variables [in Russian], Yaroslavl’ (1984), pp. 10–24.
É. S. Belinskii, “Approximation by a ‘floating’ system of exponents on classes of smooth periodic functions,” Mat. Sb., 132, No. 1, 20–27 (1987).
V. E. Maiorov, “Trigonometric widths of the Sobolev classes W p r in the space L q ,” Mat. Zametki, 40, No. 2, 161–173 (1986).
G. G. Magaril-Il’yaev, “Trigonometric widths of the Sobolev classes of functions in R n ,” Tr. Mat. Inst. Akad. Nauk SSSR, 181, 147–155 (1988).
V. N. Temlyakov, “Approximation of functions with bounded mixed derivative,” Tr. Mat. Inst. Akad. Nauk SSSR, 178, 3–112 (1986).
A. S. Romanyuk, “Kolmogorov and trigonometric widths of the Besov classes B p r , \( \theta \) of periodic functions of many variables,” Mat. Sb., 197, No. 1, 71–96 (2006).
S. A. Stasyuk, “Trigonometric widths of the classes B p Ω , \( \theta \) of periodic functions of many variables,” Ukr. Mat. Zh., 54, No. 5, 700–705 (2002); English translation: Ukr. Math. J., 54, No. 5, 852–861 (2002).
S. A. Stasyuk, “Best approximations and Kolmogorov and trigonometric widths of the classes B p Ω , \( \theta \) of periodic functions of many variables,” Ukr. Mat. Zh., 56, No. 11, 1557–1568 (2004); English translation: Ukr. Math. J., 56, No. 11, 1849–1863 (2004).
D. B. Bazarkhanov, “Estimates for some approximate characteristics of the Nikol’skii–Besov spaces of generalized mixed smoothness,” Dokl. Ros. Akad. Nauk, 426, No. 1, 11–14 (2009).
A. S. Romanyuk and V. S. Romanyuk, “Trigonometric and orthoprojection widths of the classes of periodic functions of many variables,” Ukr. Mat. Zh., 61, No. 10, 1348–1366 (2009); English translation: Ukr. Math. J., 61, No. 10, 1589–1609 (2009).
G. Akishev, “On the M-term approximations of the Besov classes,” in: Abstr. of the Internat. Conf. “Theory of Approximation of Functions and Its Applications” Dedicated to the 70th Birthday of A. I. Stepanets (1942–2007) (Kamenets-Podol’skii, 28.05–03.06, 2012), p. 12.
N. P. Korneichuk, Extremal Problems of Approximation Theory [in Russian], Nauka, Moscow (1976).
V. M. Tikhomirov, Some Problems of Approximation Theory [in Russian], Moscow University, Moscow (1976).
R. A. De Vore and V. N. Temlyakov, “Nonlinear approximation by trigonometric sums,” J. Fourier Anal. Appl., 2, No. 1, 29–48 (1995).
S. A. Stasyuk, “Best m-term trigonometric approximation for the classes B p r , \( \theta \) of functions of low smoothness,” Ukr. Mat. Zh., 62, No. 1, 104–111 (2010); English translation: Ukr. Math. J., 62, No. 1, 114–122 (2010).
A. P. Uninskii, “Inequalities in a mixed norm for trigonometric polynomials and entire functions of finite powers,” in: Imbedding Theorems and Their Applications, Proc. of the Symp. on Imbedding Theorems (Baku, 1966) [in Russian], Nauka, Moscow (1970), pp. 112–118.
É. S. Belinskii and É. M. Galeev, “On the least value of the norms of mixed derivatives of trigonometric polynomials with given number of harmonics,” Vestn. Mosk. Univ., Ser. Mat. Mekh., No. 2, 3–7 (1991).
A. Benedek and R. Panzone, “The space L p with mixed norm,” Duke Math. J., 28, No. 3, 301–324 (1961).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, No. 6, pp. 723–732, June, 2014.
Rights and permissions
About this article
Cite this article
Akishev, G. Trigonometric Widths of the Nikol’skii–Besov Classes in the Lebesgue Space with Mixed Norm. Ukr Math J 66, 807–817 (2014). https://doi.org/10.1007/s11253-014-0975-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-014-0975-y