Estimates of the best approximations of functions with a spectrum with a given majorant

  • Zh. K. Embergenova
  • M. B. Sikhov


Assessment of the best approximations of functions through its modulus of smoothness (direct theorems of approximation theory or theorems of Jackson type) and of modulus of smoothness through its best approximation in a given metric (inverse theorems of approximation theory or theorems similar to Bernstein's theorem) by orthogonal systems have been the subject of research of many generations of mathematicians. The relevance of this topic is determined by the numerical analysis, particularly with the development of computer technology. This topic is naturally linked to the further development of the tasks aiming to estimate the closest
approximation of a function, approximation of inequalities of Bernstein and Nikolskiy, and of approximation theory [1-8].


1. Temlyakov V. N. Approximation of functions with bounded mixed derivative // Trudy. – Vol.178. – M: Science, 1986. – 112 p (in Russian)
2. Pustovoitov N.N. Approximation of multidimensional functions with a given majorant of mixed moduluos of continuity. // Matem.zametki. 1999. – Vol.65, No 1. – P. 107-117 (in Russian)
3. Sikhov M.B. Some problems in the theory of multi-dimensional approximations of different metrics // Kazakh National Al-Farabi – Almaty: Kazakh University, 2010 (in Russian)
4. Sikhov M.B. On direct and inverse theorems of approximation theory with a given majorant // Analysis Mathematica. – Vol. 30, No 2. – 2004. - P. 137-146 (in Russian)
5. Sikhov M.B. Inequalities of Bernstein and Jackson – Nikolskii and evaluation norms of derivatives of the Dirichlet // Math. – Vol.80. – 2006. – P. 95-104 (in Russian)
6. Pustovoitov N.N. The ortho-diameters of multivariate periodic functions, majorant of mixed moduluos of continuity which contains as power and logarithmic factors // Anal.Math. – 2008. – Vol. 34. – P.187-224 (in Russian)
7. Pustovoitov N.N. On the approximation of periodic functions by linear methods of the classes // Matem.sbornik. – 2012. – Vol. 203, No 1. – P. 91-113 (in Russian)
8. Sikhov M.B. Approximation of functions of several variables with a given majorant space Besov // Mathematical journal. – 2002. – Vol.2, No 2. – P. 95-100 (in Russian)
How to Cite
EMBERGENOVA, Zh. K.; SIKHOV, M. B.. Estimates of the best approximations of functions with a spectrum with a given majorant. International Journal of Mathematics and Physics, [S.l.], v. 7, n. 1, p. 21-26, june 2016. ISSN 2409-5508. Available at: <>. Date accessed: 21 may 2018. doi:


the best approximation, assessment of the best approximation, modulus of continuity