On M-term approximations of the Nikol’skii - Besov class

Year 2016, Volume: 45 Issue: 2, 297 - 310, 01.04.2016

G. Akishev

Abstract

In this paper, we consider a Lebesgue space with a mixed norm of
periodic functions of many variables. We obtain the exact estimation
of the best M-term approximations of Nikol’skii’s and Besov’s classes
in the Lebesgue space with the mixed norm.

Keywords

Lebesgue space, Nikol’skii - Besov class, approximation

References

• Akishev, G. On the exact estimations of the best M-term approximation of Besov’s class, Siberian Electronic Mathematical Reports 7, 255 – 274, 2010.
• Akishev, G. On the order of the M-term approximation classes in Lorentz spaces, Matematical Journal. Almaty 11, (1), 5 - 29, 2011.
• Akishev, G. On the M-term approximation Besov’s classes, International Conference “Theory of approximation of functions and its applications” dedicated to the 70-th anniversary of corresponding member of National Academy of Ukraine, professor A.I. Stepanets (1942 – 2007) May 28 - June 3, Ukraine, Kamianets-Podilsky, p. 12 , 2012.
• Amanov, T.I. Spaces of differentiable functions with dominant mixed derivative, Alma-Ata, 1976.
• Bazarkhanov, D.B. Estimates for certain approximation characteristics of Nikol’skii-Besov spaces with generalized mixed smoothness, Doklady Ros. Akade. Nauk 426, (1), 11 – 14, 2009.
• Belinsky, E.S. Approximation by a ’floating’ system of exponents on the classes of smooth periodic functions, Mat. sb. 132, (1), 20 – 27, 1987.
• Besov, O.V. Investigation of a family of functional spaces in connection with the theorems of imbedding and extension, Trudy Mat. Inst. Akad. Nauk SSSR 60, 42 - 61, 1961.
• De Vore, R.A. Nonlinear approximation, Acta Numerica 7, 51 – 150, 1998.
• De Vore, R.A. and Temlyakov, V.N. Nonlinear approximation by trigonometric sums, Jour. of Fourier Analysis and Applications 2, (1), 29–48, 1995.
• Dinh Dung On asymptotic order of n-term approximations and non-linear - n widths, Vietnam Journal Math. 27, (4), 363 - 367, 1999.
• Duan, L. The best m-term approximations on generalized Besov’s classes MBΩ q,θ with regard to orthogonal dictionaries, Journal Approximation Theory 162, 1964 – 1981, 2010.
• Hansen, M. and Sickel, W. Best m-term approximation and Lizorkin-Triebel spaces, Journal Approximation Theory 163, 923 – 954, 2011.
• Ismagilov, R.S. Widths of sets in linear normed spaces and the approximation of functions by trigonometric polynomials, Uspehi mathem. nauk 29, (3), 161 – 178, 1974.
• Kashin, B.S. Approximation properties of complete orthnormal systems, Trudy Mat. Inst. Steklov 172, 187 – 201, 1985.
• Kashin, B.S., and Sahakyan, A.A. Orthogonal series, Nauka, Moscow, 1984.
• Lizorkin, P.I. Generalized Holder spaces B (r) p,θ and their correlation with the Sobolev spaces L (r) p , Sibirsk. Mat. Zh. 9, 1127 - 1152, 1968.
• Maiorov, V.E. On linear widths of Sobolev classes and chains of extremal subspaces, Mat. sb. 113, (3), 437 – 463, 1980.
• Nikol’skii, S. M. Approximation of classes of functions of several variables and embeding theorems, Nauka, Moscow, 1977.
• Romanyuk, A.S. On the best M-term trigonometric approximations for the Besov classes of periodic functions of many variables, Izv. Ros. Akad. Nauk, Ser. Mat. 67, (2), 61 – 100, 2003.
• Stasyuk, S.A. Best m-term trigonometric approximation for the classes Br p,θ of functions of low smoothness, Ukrain. Mathem. Journal 62, (1), 114 –122, 2010.
• Stasyuk, S.A. Best m-term trigonometric approximation of periodic functions of several variables from Nikol’skii - Besov classes for small smoothness, Journal of Approximation Theory 177 , 1 - 16, 2014.
• Stechkin, S.B. On the absolute convergence of orthogonal series, Doklad. Akadem. Nauk SSSR 102, (2) , 37 – 40, 1955.
• Temlyakov, V. N. Nonlinear methods approximation, Foundations of Computational Mathematics 3, 33–107, 2003.
• Uninskii, A.P. Inequalities for entire functions and trigonometric polynomials with mixed norm, Proceeding of the All-Union Symposium on embedding theorems, ( Baku - 1966) - M.: Nauka, 112 - 118, 1970.